EMM3808
MECHANICAL ENGINEERING LABORATORY II
(LAB MANUAL)
DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING
FACULTY OF ENGINEERING
UNIVERSITI PUTRA MALAYSIA
Lecturer(s):
(Group 1) Dr. Mohd Zuhri Mohamed Yusoff
(Group 2) Assoc. Prof. Dr. Edi Syams Zainudin
Assistant Engineer(s):
Nor Mahayon Mohd Mahayuddin
Khairuddin Haikal Noor Azhar
Noor Azmi Ismail
Mohd Saiful Azuar Md. Isa
Muhammad Wildan Ilyas Mohamed Ghazali
Mazrul Hisham Mustafa Kamal
TITLE
PAGE
LAB 1
HYDROSTATIC PRESSURE
1
LAB 2
METACENTRIC HEIGHT
8
LAB 3
VISCOCITY & DENSITY
13
LAB 4
BERNOULLI'S THEOREM
17
LAB 5
IMPACT OF JET
22
LAB 6
TORSION OF NON-CIRCULAR SHAFT
27
LAB 7
STATICALLY INDETERMINATE BEAM TEST
32
LAB 8
THERMAL STRESS ANALYSIS & BUCKLING OF STRUT
35
LAB 9
THIN CYLINDER
48
LAB 10
THERMAL EXPANSION FOR SOLIDS & LIQUIDS
53
LAB 11
HEAT CAPACITY (SOLIDS, LIQUIDS & AIR)
55
LAB 12
TEMPERATURE MEASUREMENT
68
LAB 1: HYDROSTATIC PRESSURE
Introduction
The effect of hydrostatic pressure is of major significance in many areas of engineering, such as
shipbuilding, the construction dykes, weirs and locks, and in sanitary and building services
engineering.
Objective
To determine the centre of pressure on a partially submerged plane surface
Theoretical Background
The hydrostatic pressure of liquids is the “gravitational pressure” :
phyd =  .g.t
 = Density of water
g
= Accelaration due to gravity ( g
t
= Distance from liquid surface
= 9.81m / s 2 )
Determining the Center of Pressure
A linear pressure profile is acting on the active surface shown in Figure 1, because the hydrostatic
pressure rises proportional to the depth t. The resultant force Fp is therefore not applied at the
centre of force C of the active surface, but always slightly below it, at the so centre of pressure D.
To determine the distance of e of the centre of pressure from the planar centre of force, the
following model demonstration is used:
Figure 1: Centre of Pressure D
1
Imagine an area A in front of the active surface, formed by the height h and the pressure profile of
the hydrostatic pressure p1 − p 2 . This area is in the form of trapezium. The centre of pressure D
lies on the extension of the planar centre of force of this area A. A can be broken down into partial
areas A1 and A2 . The respective planar centres of forceare identified by black dots.
A balance of moments between areas is then established around the point O1 in order to find the
common planar centre of force (dynamic effect in direction Fp ) see Figure 2:
Figure 2: Planar Centre of Gravity
h 
M (O1 ) = 0 : A. + e  = A1. h + A2 . 2h
2
3
2 
Where
A1 = p1.h
p2 − p1
.h
2
A = A1 + A2
A2 =
and
The result is
e=
1 p2 − p1
h.
6 p2 + p1
With the hydrostatic pressure
h

p2 = g cos  . yc +  and
2

h

p1 = g cos  . yc −  the result is
2

e=
1 h2
.
12 yc
2
e is the distance of the centre of pressure from the planar centre of force or the active surface which
we are looking for.
At the water level s, below the 100mm mark, the height of the active surface changes with the
water level. If the water level is above that mark, the height of the active surface is always 100mm,
see Figure 3. Meaning:
s = Water lavel
e = Distance of centre of pressure D from the planar centre of
force C of the active surface
I D = Distance to centre of motion of the unit:
Figure 3
For a water level s<100mm (pressure has a triangular profile):
1
e = .s
6
1
I D = 200mm − .s
3
For a water level s>100mm (pressure has a trapezoidal
profile) see Figure 4:
1 (100mm)
e= .
12 s − 50mm
I D = 150mm + e
2
When the water vessel is at a tilt, a triangular pressure profile is
produced when the water level is below
sh ; above that level a
trapezoidal profile is produced, see Figure 5. Measured values:
Figure 4
s = Water level reading
α = Tilt angle of vassel
st = Water level at lowest point of vessel
sh = Water level of active surface at rim
e = Position of centre of pressure
h = Height of active surface
Figure 5
I D = Distance between centre of pressure
3
For a water level s< sh a triangular profile as follow applies:
s − st
cos 
1
e = .h
6
h=
1
I D = 200mm − .h
3
For water level s> sh a trapezoidal profile as follows applies:
(100mm)
1
e= .
s
−
st
12
− 50mm
cos 
I D = 150mm + e
2
Apparatus
•
Hydrostatic Pressure Apparatus HM 150.05.
1.
2.
3.
4.
5.
6.
7.
8.
Figure 6: Hydrostatic Pressure Apparatus HM 150.05.
4
Water vessel
Detent
Slider
Stop pin
Water level scale
Rider
Weights
Handles
Procedure
Experiment 1: Centre of Pressure with Vertical Positioning of the Water Vessel
1. Set the water vessel (1) to an angle  = 0o using the detent (2) as shown in Figure 7
2. Counterbalace the unit with a rotating slider (3). The stop pin (4) must be precisely in the
middle of the hole for this.
3. Mount the rider (6), set the lever arm on the scale I=150mm see Figure 8
4. Increase the appended weights (7) in increments of 2N-8N
5. Top up with water until the unit is balanced (stop pin (4) at centre of hole)
6. Repeat procedure 3 in increments of I=160mm,170mm,180mm and 190mm.
Figure 7
Figure 8
Experiment 2: Centre of Pressure with Water Vessel Tilted
1. Set an angle α = 30° and counterbalance
the water vessel as described under
procedure 1&2 on Experiment 1
2. Enter the characteristic values in prepared
worksheet for the lowest water level st and
highest water level
Figure 9
sh of the active surface
see Figure 9
3. Perform a measurement as described under
procedure 3 until procedure 6 on
Experiment 1.
4. Repeat procedure 1 with an angle α = 60°
& 90°.
5
Results
Experiment 1: Centre of Pressure with Vertical Positioning of the Water Vessel
Angle α[°]
0
Lever arm I [mm]
Appended weight
Water level reading
s[mm]
FG
[N]
150
2
150
4
150
6
150
8
Distance between
centre of pressure I D
[mm]
Experiment 2: Centre of Pressure with Water Vessel Tilted
Angle α[°]
Lowest Water Level s
t
Highest Water Level
[mmWC]
sh
[mmWC]
30°
Lever arm I [mm]
Appended weight
Water level reading
s[mm]
FG
[N]
150
2
150
4
150
6
150
8
Distance between
centre of pressure I D
[mm]
6
Discussion and Conclusions
1. Define the water level reading, s, and distance between center of pressure, I D , for the various
lever arm length (at an angle 0°) in your spread sheet and explain the results.
2. Define water level reading s and distance between center of pressure I D for the various lever
arm length (at various angles) in your spread sheet and explain the results.
3. Give reasons for the discrepancies, if any, between the measure and predicted values of the
above expressions for the graph parameters.
4. Conclude your findings.
7
LAB 2: METACENTRIC HEIGHT
Introduction
The position of the so-called metacentre, the metacentric height, is of crucial significance to the
stability of a floating body. The metacentric height is an essential factor when assessing the
stability of a ship in waves.
Objective
To study the stability of a floating body experimentally and compare with the calculated
stability.
Theoretical Background
Buoyancy
A body floats in a liquid if the buoyancy of the fully
immersed body is greater than its weight. It will only
sink into the liquid until the buoyancy FA corresponds
exactly to its dead weight FG. The buoyancy is then the
weight of the water displaced by the body. The centre of
gravity of the displaced water mass is referred to as the
centre of buoyancy A. the centre of gravity of the body
is known as the centre of mass S.
Stability of Floating Body
For a floating body to be stable, buoyancy FA and dead
weight FG must have the same line of action and be
equal and opposite. Stability does not necessarily
demand that the centre of mass S be below the centre of
buoyancy A.
Of far greater importance is the existence of a
stabilizing, resetting moment in the event of deflection
or heel α out of the equilibrium position. Dead weight
8
FG and buoyancy FA then form a force couple with
distance b, which provides a righting moment. This
distance or the distance between the centre of gravity
and the point of intersection of line of action of
buoyancy and gravity axis, is a measure of stability.
This point of intersection is referred to as the
metacentre M and the distance between the centre of
gravity and the metacentre is called the metacentric
height zm.
The following conditions then apply to stable floating:
•
Stable floating of a body occurs when the
metacentric height Zm is positive, i.e. the metacentre
M is above the centre of gravity S. Zm > 0.
• Unstable floating of a body occurs when the
metacentric height Zm is negative, i.e. the metacentre
M is below the centre of gravity S. Zm < 0.
Determination of Metacentre Position
The position of the metacentre is not governed by the
position of the centre of gravity. It merely depends on
the shape of the portion of the body under water and the
displacement. There are two methods of determining
the position by way of experiment.
In the first method, the centre of gravity is laterally
shifted by a certain constant distance xS using an
additional weight, thus causing heeling to occur.
Further vertical shifting of the centre of gravity alters
the heel α. A stability gradient formed from the
derivation
𝑑𝑥𝑠
𝑑𝛼
is then defined. The stability gradient
decreases as the vertical centre of gravity position
9
approaches the metacentre. If centre of gravity
position and metacentre coincide, the stability
gradient is equal to zero and the system is
metastable.
This problem is most easily solved using a graph as
in the figure. The vertical centre of gravity position
is plotted versus the stability gradient. A curve is
drawn through the measurement points and
extended as far as the vertical axis
𝑑𝑥𝑠
𝑑𝛼
= 0. The
point of intersection with the vertical axis then
gives the position of the metacentre.
With the second method of determining the metacentre, it is assumed that, given a stable heel
position, dead weight FG and buoyancy FA have one line of action. The point of intersection of this
line of action with the central axis gives the metacentre M. The heel angle α and the lateral
displacement of the centre of gravity XS yield the following for the metacentric height Zm.
𝑧𝑚 = 𝑥𝑠 cot 𝛼
Calculation of Centre of Gravity Position
The first step is to determine the position of the overall centre of gravity xs, zs from the set
position of the sliding weights. The horizontal position is referenced to the centre line:
10
𝑥𝑠 =
𝑚ℎ 𝑥
= 0.055𝑥
𝑚 + 𝑚𝑣 + 𝑚ℎ
The vertical position is referenced to the underside
of the floating body:
𝑧𝑠 =
𝑚𝑣 𝑧 + (𝑚 + 𝑚ℎ )𝑧𝑔
= 5.364 + 0.156𝑧
𝑚 + 𝑚𝑣 + 𝑚ℎ
Stability gradient:
𝑑𝑋𝑠 𝑋𝑠
=
𝑑𝛼
𝛼
Apparatus
Hydraulic Bench (Armfield F1-10), Metacentric Height Apparatus (Armfield F1-14), Ruler,
Weighing Machine
11
Procedure
1.
2.
3.
4.
Set horizontal sliding weight to position x = 8 cm
Move vertical sliding weight to bottom position
Fill tank provided with water and insert floating body
Gradually raise vertical sliding weight and read off angle on heel indicator. Read off height
of sliding weight at top edge of weight and enter in table together with angle.
Results
Position of horizontal sliding weight x = 8 cm
Height of vertical
sliding weight z
Angle α
Horizontal position of centre of gravity xs =
cm
Height of vertical
sliding weight, z
Centre of gravity
position, zs
Angle, α
𝑑𝑋𝑠
𝑑𝛼
Plot graph vertical centre of gravity position zs versus stability gradient
𝑑𝑋𝑠
𝑑𝛼
Discussion and Conclusions
1. Does the position of the metacentre depend on the position of the C of G?
2. Does the metacentre height vary with angle of heel?
3. What are the effects of changing the position of G on the position of the metacentre?
4. Why the values of GMat lowest level of 𝜃 are likely to be less accurate?
5. Conclude your findings.
12
LAB 3: VISCOCITY & DENSITY
Introduction
The results of viscosity measurement with the vibro viscometer apparatus concerning several
types of fluid will be determined hereinafter. It is purposely used to determine both Newtonian
and non-Newtonian fluid.
Objective
To identify Newtonian and non-Newtonian fluids and its viscosity profile against time and
temperature.
Theoretical background
Viscosity is a measure of resistance of a fluid to deform under shear stress. It is commonly
perceived as ‘thickness’, or resistance to flow. Viscosity describes a fluid’s internal resistance to
flow and may be thought of as a measure of fluid resistance. All real fluids have some resistance
to shear stress or known as viscous fluids, but a fluid which has no resistance to shear stress is
known as an ideal fluid or inviscid fluid.
y
Board B
N
A viscous fluid can be presented as in Figure 1, where
the two plates, board A and board B are placed
parallel, filled with a fluid in between with distance,
y0. Board A is fixed while board B is being moved at a
constant speed of V0. If the fluid between board A and
B also in parallel motion with fluid velocity of V and
has a steady flow, this is called Couette flow.
M
Board A
Figure 1. Couette flow fluid
The slope connecting M-N can be calculated using Equation 1, where it equals to the increased
quantity of the velocity per unit distance.it is also called as shearing rate or velocity gradient.
𝐷=
𝑑𝑉
𝑑𝑦
(1)
Because there are differences in the velocities, an internal frictional force will develop between
them. The frictional force applying to the unit area of the plane of the flow between board A and
B is called a shear stress, τ, which is proportional to D as shown in Equation 2.
τ = ηD
(2)
13
Eq. 2 represents the law known as Newton’s law of viscosity, µ is denote as dynamic viscosity.
Dividing µ with the fluid density, ρ will give another equation called kinematic viscosity, υ.
υ = µ/ρ
(3)
Fluids that follow the Newton’s law, whose η at specific temperature is constant called a
Newtonian fluid such as water, alcohol, sugar solution and etc. if the µ is not constant, it is called
non-Newtonian fluid such as paint, tomato ketchup and etc.
Apparatus
Vibro viscometer main unit and display unit.
14
Procedure
1. Preparing sample:
a. Pour the sample into the cup until its surface reaches between the level gauges, which are
between 35 to 45 mL.
b. Attach the cup on the table along the guides
c. Make sure the protector is in the position as shown in Figure 1
d. Raise the lever to release the sensor plates
e. Hold the front side of the sensor unit the pinch the grips and slowly lower the sensor
plates above the sample surface.
f. Lower the lever to secure the sensor plate
g. Turn the knob to adjust the sample surface to the center of the narrow part of the sensor
plate as shown in Figure 2.
2. Measurement:
a. Press the START key to display the measurement values. After ±15 seconds the first
measurement values will be displayed.
b. Press HOLD key to temporarily freeze the display and write down the value.
c. Take reading every 30 seconds for 10 minutes then press STOP key.
3. Repeat step 1 and 2 for other samples.
Results
Plot the graph for viscosity against temperature and viscosity against time (in minute).
Time (minute)
Temperature (°C)
Viscosity, η
2
2.5
4
4.5
6
6.5
8
8.5
10
10.5
15
Discussion and Conclusions
1. Identify the Newtonian and non-Newtonian fluid from the plotted graphs? Why?
2. Find another 3 Newtonian and non-Newtonian fluid with its viscosity.
3. State one engineering sample that using fluid as its medium and described the important of
fluid viscosity in this application.
4. Conclude your findings.
16
LAB 4: BERNOULLI’S THEOREM
Introduction
The Venturi meter is a device which has been used over many years for measuring the discharge
or flow along a pipe. The fluid flowing in the pipe is led through a contraction section to a throat,
which has a smaller cross-sectional area than the pipe, so that the velocity of the fluid through the
throat is higher than that in the pipe. This increase of velocity is accompanied by a fall in pressure,
the magnitude of which depends on the rate of flow, so that by measuring the pressure drop, the
discharge may be calculated. Beyond the throat the fluid is decelerated in a pipe of slowly
diverging section (sometimes referred to as a diffuser) in order to recover as much of the kinetic
energy as possible.
Objective
• To determine the discharge coefficient (Cd) of venture meter
• To measure the volume flow rate using venture meter.
Theoretical Background
Pressure differences can be used to determine the volume flow rate for any particular
configuration. Venturi meter used this pressure differences for the measurement of flow in
pipelines. The venturi meter consists of a venturi tube and differential pressure gauge. The tube
has a converging portion, a throat and a diverging portion as shown in Fig 1. The pressure
difference from which the volume rate of flow can be determined is measured between the inlet
section 1 and the throat section 2.
Figure 1. Venturi Tube
17
Theoretically according to Douglas et. al. (1992), assuming that there are no energy loss and by
applying Bernoulli’s equation to section 1 and 2:
𝑃1 𝑣12
𝑃2 𝑣22
𝑍1 +
+
= 𝑍2 +
+
𝜌𝑔 2𝑔
𝜌𝑔 2𝑔
𝑣22 − 𝑣12 = 2𝑔 [(
𝑃1 − 𝑃2
) + (𝑧1 − 𝑧2 )]
𝜌𝑔
(1)
Where,
z1 – z2 is the difference of level
v1,2 represents the velocities at 1 and 2
P1,2 denote the pressure at 1 and 2
g is the gravity = 9.81 m/s2
ρ is the fluid density, for water = 1000 kg/m3
For continuous flow,
A1v1 = A2v2
Or
𝑣2 = (
𝐴1
)𝑣
𝐴2 1
(2)
Where A1 is the inlet area = 5.309x10-4 m2 and A2 is the throat area = 2.011x10-4 m2.
Substituting Eq. 2 into 1,
𝑣12 [(
𝐴1 2
𝑃1 − 𝑃2
) − 1] = 2𝑔 [(
) + (𝑧1 − 𝑧2 )]
𝐴2
𝜌𝑔
Since the venturi meter is in horizontal position, so z1 = z2 and the pressure differences,
𝑃1 −𝑃2
𝜌𝑔
between section 1 and 2 is represent by manometer tube 1 (h1) and 3 (h3) or h1 - h3. Therefore, the
equation can be written as follow:
𝐴1
𝑣12 [( )
𝐴2
2
− 1] = 2𝑔(ℎ1 − ℎ3 )
(3)
So, velocity at section 1 can be determined using Eq. 4
18
𝑣1 =
2𝑔(ℎ1 − ℎ3 )
(4)
√ 𝐴1 2
[( ) − 1]
𝐴2
The theoretical volume flow rate is QT = A1v1 and by substituting Eq. 4 into the equation. It
becomes,
𝑄𝑇 = 𝐴1
2𝑔(ℎ1 − ℎ3 )
(5)
√ 𝐴1 2
[( ) − 1]
𝐴2
However, in practice, some loss of energy will occur between section 1 and 2. The value of Q in
Eq. 5 will be slightly greater than actual value QA. A coefficient of discharge, Cd is introduced to
determine the actual value using Eq. 6
QA = CdQT
(6)
Apparatus
Hydraulic bench, stop watch and venturi meter
Figure 2. Arrangement of Venturi Meter apparatus
19
Procedure
1. Adjust the discharge valve to the maximum measurable flow rate of the venturi meter (this can
be achieved when tube 1 and 3 give the maximum water head difference).
2. After the fluid level stabilized, note the fluid level of the manometers.
3. Measure the volume flow rate of the fluid using volumetric method by closing the water outlet
in the hydraulic bench. Determine the time of 5 liters fluid volume using stop watch.
4. Repeat step 1 to 3 for three other decreasing flow by regulating the venturi discharge valve.
Results
Volume (liter)
Time (s)
Table 1. Manometer and actual water volume data
Manometer reading, h (mm)
1
2
3
4
5
6
5
5
5
5
1.
2.
3.
4.
5.
From Table 1, obtain the experiment volume flow rate (QE) from the volumetric method.
Calculate the theoretical volume rate using the manometer reading h1 and h3 using Eq. 5
Plot a graph of QE against QT and determine the discharge coefficient, Cd from the slope.
Use Cd to obtain actual volume rate using Eq. 6
Find the errors between QE and QA.
QE
Table 2. Volume flow rate data.
h1-h3
QT
QA
Errors
𝑄𝐸 − 𝑄𝐴
𝑥100%
𝑄𝐸
20
Discussion and Conclusions
1. Discuss the data obtained from Table 1 and 2.
2. What is the theoretical range of Cd?
3. Is the experimental Cd obtained within the range? Why?
4. Can the experimental results be accepted or rejected? Why?
21
LAB 5: IMPACT OF JET
Introduction
One way of producing mechanical work from fluid under pressure is to use the pressure to
accelerate the fluid to a high velocity in a jet. The jet is directed on to the vanes of a turbine wheel,
which is rotated by the force generated on the vanes due to the momentum change or impulse that
takes place as the jet strikes the vanes. Water turbines working on this impulse principle have been
constructed with outputs of the order of 100 000kW with efficiencies greater than 90%.
In this experiment, the force generated by a jet of water as it strikes a flat plate or hemispherical
cup may be measured and compared with the momentum flow rate in the jet.
Objective
To investigate the validity of theoretical expressions for the force exerted by a jet on targets of
various shapes.
Theoretical Background
Consider a vane symmetrical about the x-axis as shown in Figure 1. A jet of fluid flowing at the
rate of 𝑚̇ kg/s along the x-axis with the velocity 𝑢0 m/s strikes the vane and is deflected by it
through angle β, so that the fluid leaves the vane with the velocity 𝑢1 m/s inclined at an angle β to
the x-axis. Changes in elevation and piezometric pressure in the jet from striking the vane to
leaving it are neglected.
Figure 1. Vane symmetrical about x-axis
22
Momentum enters the system in the x-direction at a rate of:
𝑚̇𝑢0 (kg m/s2)
Momentum leaves the system in the same direction at the rate of:
̇
𝑚𝑢1 𝑐𝑜𝑠𝛽
(kg/ms2)
The force on the vane in the direction is equal to the rate of change of momentum change.
Therefore:
𝐹 = 𝑚̇(𝑢0 − 𝑢1 𝑐𝑜𝑠𝛽)
(Newton)
Ideally, jets are ‘isotachtic’, or constant velocity so that u0 = u1. Therefore:
𝐹 = 𝑚̇𝑢0 (1 − 𝑐𝑜𝑠𝛽)
(Newton)
Shape
Table 1. Types of vane
β
F
|
90°
𝑚̇𝛽𝑢0
>
120°
1.5 𝑚̇𝑢0
)
180°
2 𝑚̇𝑢0
\
30°
0.87 𝑚̇𝑢0
23
Apparatus
Hydraulic bench H1D, H8 Impact of Jet Apparatus, Vernier caliper, stop watch.
Figure 2. Diagrammatic arrangement of apparatus.
24
Procedure
1. Level the apparatus and set the lever to balance position (as indicated by tally) with the jockey
weight at its zero position.
2. Admit the water through the bench supply valve.
3. Increased the flow rate to the maximum and take note the position of the jockey weight which
restores the lever to the balance position. Weight the discharge in the weighing tank.
4. Take about eight readings with roughly equally space position of the jockey weight by
decreasing the flow rate from the bench (Adjust weight of water collected to ensure discharge
over 60 seconds).
5. Repeat the experiment using hemispherical cup, conical plate and angled plate in turn.
6. Take note of the diameter of the nozzle, the height of the vane above the tip of the nozzle when
the lever is balanced, the distance between the centre of the vane and the pivot of the lever as
well as the jockey weight.
Results
Density of water
Diameter of nozzle
Cross-sectional area of nozzle, A
Mass of jockey weight
Distance from centre of vane to
pivot of lever
Quantity
(kg)
t
y
(s)
(m)
: ____________
: ____________
: ____________
: ____________
: ____________
Table 2. Results
u
𝒎̇
(m/s)
(kg/s)
𝒖𝟎
𝒎̇𝒖𝟎
F
(m/s)
(N)
(N)
**Repeat table for different cup.
Note: Quantity refers to the actual weight of collected water.
1. Compute the force developed on vane of various shape.
2. Compute the rate of delivery momentum, 𝑚̇𝑢0 (kg m/s2)
3. Plot graph force on vane for various shapes versus rate of delivery momentum for both
experiment and theoretical methods.
25
Discussion and Conclusions
1.
2.
3.
4.
5.
What suggestions would you propose to improve the apparatus?
What would be the effect on the calculated value of the efficiency of the following errors
measurement:
a. Jockey weight in error by 1 g;
b. Distance from centre of vane to pivot of lever in error by 1 mm.
The ideal flow model has assumed that the jet has a uniform velocity distribution and is
constant around the vane. A real jet has a velocity distribution from zero at the edge to a
maximum in the centre. It is not actually parabolic, but consider the effect that this would
have on the force. The average value is still u0. Real jet also spread and slow down. What
would be the effect of a jet 10% greater area and 10% slower at the vane compared with that
emerging from the apparatus?
If the cone and the hemisphere faced the other way, i.e. the open section away from the jet,
what would the ideal force be? Why does momentum theory not predict the actual results?
Conclude your result.
26
LAB 6: TORSION OF NON-CIRCULAR SHAFT
Objective
The objectives of this experiment are:
1. To develop the relationship between torque T and shear stress τ with angle of twist θ.
2. To determine the modulus of rigidity G of the material.
3. To determine the maximum shear stress at the elastic limit and at failure of the
material.
Theory Background
General
The torsion test differs from a tensile test in that the former has a stress gradient across the crosssection of the specimen. Thus, at the limit of the elastic range, yielding will first occur in the
outermost fibers whilst the core is still elastic. In a tensile test, yielding occurs relatively evenly
throughout the bar.
As the specimen is twisted further into the plastic region, a greater proportion of the cross section
yields until there is a plastic zone through to the centre of the bar. A typical torque–twist diagram
(Figure 1) is very similar to a load–extension diagram from a tensile test.
Figure 1. Torque-twist diagram
Torsion of Non-Circular Section
Shear stress cannot act in a direction normal to a free surface. It follows that at a corner the shear
stress is zero.
A way of visualizing the shear stress over a section is found by the use of the elastic membrane
analogy. Imagine a hollow section with the same dimension covered in a thin elastic membrane.
The inside is then pressurized so that the membrane expands. The shape obtained for a rectangle
is shown as Figure 2.
27
Figure 2. Membrane expands for rectangular shape
The gradient of the membrane represents the shear stress and this is zero in the corners, zero in the
centre and maximum at the centre of the longest edge. This theory also works with other shapes
such as T-section like shown in Figure 3.
Figure 3. Membrane expands for T-shape
Rectangular Section
The maximum shear stress in a rectangular section is:
And it occurs at the mid-point of the longest edge. The angle of twist is:
Where L is the length of the shaft. α and β are figures that depend on the ratio of the dimensions
and are given by the following formula.
Figure 4. Rectangular Section
28
Note that as the ratio n increases the values of α and β tend towards a value of 1/3 and for ratios
larger than 10 this figure is used.
Some sources use a constant called apparent polar second moment of area K which for rectangle
is:
Specimens and Equipment
1. Torsion testing machine – Norwood 50 Nm
2. Vernier caliper
3. Torsion specimens: steel, mild steel, aluminium
29
Figure 5. Norwood 50 Nm
Procedures
1. Measure the initial length and initial gauge length diameter of the specimen.
2. The specimen between the loading device and the torque-measurement unit into the straining
hexagon sockets.
3. Turn the hand-wheel as required aligning the specimen.
4. Slide the tailstock unit so that the specimen is fully into the hexagon sockets.
5. Ensure that there is no preload on the specimen.
6. Zero the pointer on the zero-degree point on the protractor scale.
7. Adjust the digital torque meter reads zero.
8. Turn the hand-wheel clockwise slowly to load the specimen. Turn it only for a defined angle
increment.
9. Read the torque value from the digital torque meter and notice it together with the indicated
angle of twist.
10. Continue the process of steps 8 and 9 until fracture occurs.
11. Repeat the experiment for other specimens.
30
Results
1. Specimen Dimensions:
2. Angle of Twist and Torque.
3. Plot the torque T versus angle of twist θ graph for the tested specimen. From the graph,
calculate the modulus of rigidity G, and determine the slope of the elastic part.
4. Calculate the shear stress τ at the proportionality.
5. Plot the shear stress τ against angle of twist θ graph.
6. Plot the shear stress τ against shear strain γ graph. Tabulate the following values and show
them on the τ– γ curves
i. Proportional limit shear stress in torsion
ii. Shear modulus of elasticity
iii. Shear stress
Discussion
1. Compare and discuss on the experimental results with the theory.
2. Discuss on the mechanical properties of the tested specimens in shear.
3. Discuss on the factors that can be affected to the experimental result.
Conclusion
1. Give an overall conclusion based on the obtained experimental results.
2. Conclude on the applications of the experiment.
31
LAB 7: STATICALLY INDETERMINATE BEAM TEST
Introduction
A beam is a structural element that carries load primarily in bending (flexure). Beams generally
carry vertical gravitational forces but can also be used to carry horizontal loads. Beams are
characterized by their profile (the shape of their cross-section), their length, and their material.
Beams carry their loading to other elements or supports. In order to be able to analyze a structure
it is necessary to be clear about the forces that can be resisted at each support.
This manual contains some fundamental theory for understanding the experiment, description of
the apparatus and experimental procedure to examine the supports reaction of the beam in addition
to verify the highest deflection location.
Objectives
The objectives of this experiment are:
1. To determine the reaction of a two-span continuous beam.
2. To find the maximum location of deflection.
Theory Background
Typical reaction at the support of a continuous beam is as shown below.
Figure 1. Description of beam apparatus
Specimen and Equipment
1. A support frame
2. Support reaction pier (3 units)
3. Load hangers
4. Specimen (Steel)
5. Measuring tape
6. Set of weights
32
Figure 2. Experimental beam set-up
Procedures
1. Switch on the display unit to warm up the unit.
2. Clamped the reaction piers to the support frame using the plate and bolt supplied.
3. Place the beam specimen between the two cylindrical pieces of each support. Tightened the
two screws at the top of each support with your fingers.
4. Fix the load hanger at the position where the beam is to be loaded.
5. Connect the load cell from the support pier to the display unit, each load cell occupying one
terminal on the display.
6. Beginning the channel 1 record the initial reading for each channel.
7. Place the suitable load on the load hanger and note the reading of each load cell. This represents
the reaction at each pier.
8. Increase the load on the load hanger at suitable increments and for each increment record the
pier reaction.
Results
1. Show all the measurements of beam.
i. Beam length L [mm]
ii. Beam width b [mm]
iii. Beam thickness h [mm]
iv. Beam working length l [mm]
2. Draw the beam and indicate the position and direction of load including Shear Force and
Bending Moment Diagram. Identify the maximum location of deflection.
3. Derive the reactions at the supports
4. Using the tabulated data:
i. Plot the graph of reaction against load for each support.
ii. Draw the best-fit-curve through the plotted points.
iii. Using the slope of the graph, calculate the percentage of error between the experimental
and theoretical reaction
33
Table 1. Experimental result of reaction and deflection for beam with multiple supports
Discussion
1. If the material of the beam is changed from steel to aluminum, how does this affect the support
reaction? Give reasons for your answer.
2. If the thinner beam is used, how does this affect the support reaction? Give reasons from your
answer.
3. How does the experimental reactions compare with the theoretical?
4. State the possible factor that might have influenced your results and possible means of
overcoming it.
Conclusion
Give an overall conclusion based on the obtained experimental results.
34
LAB 8: THERMAL STRESS ANALYSIS & BUCKLING OF STRUT
A) THERMAL STRESS ANALYSIS
Introduction
Thermal stress effects can be simulated by coupling a heat transfer analysis (steady-state or
transient) and a structural analysis (static stress with linear or nonlinear material models or
Mechanical Event Simulation [MES]). The process consists of two basic steps:
1. A heat transfer analysis is performed to determine the temperature distribution; and
2. The temperature results are directly input as loads in a structural analysis to determine the
stress and displacement caused by the temperature loads.
Objective
The objectives of this experiment are:
1. To measure the change in length of test specimen.
2. To determine the thermal expansion coefficient of different test specimen.
Theory Background
Thermal stress is caused by the temperature change on a material. This stress is divided into two
categories; thermal expansion and thermal contraction where these phenomena cannot occur freely
in all directions because of geometry, external constraints or the existence of temperature gradient.
Before the effect of thermal stress, the material heated undergoes deformation process (thermal
deformation), as the temperature of material increases, the vibration of its atoms/molecules
increase in speed; thus, the molecule bond is being stretched which causes the material to expand.
On the other hand, the thermal energy decreased, the material will shrink or contract.
Thermal stress defined as the equation below:
We can also define the material’s stress in term of strain.
For a solid material with significant length, sch as rods or cables, the amount of thermal expansion
or contraction can be describe as strain εth:
Where, LO is the initial length before the change of temperature, and Lt is the final length recorded
after the change of temperature.
35
For most of the materials, thermal expansion is related directly with temperature due to the kinetic
motion of molecules:
Where,
α Is the linear coefficient of expansion for the material, and is the fraction change in length per
degree change in temperature.
L represents the initial length of the rod, and 3.
Lt is the final length recorded after the change of temperature.
∆T is the change in temperature (positive value is we have thermal expansion, and negative for
thermal contraction).
Description of the Thermal Stress Apparatus
This apparatus is designed for students to measure the material thermal expansion/stress for
different types of material. The unit comprise of an epoxy coated steel frame, temperature sensor,
temperature meter, dial gauge, control panel, heater and three different type of material (steel,
stainless steel, and copper).
A simple way to determine the amount of stress is to let the material expand freely due to thermal
expansion, and then compress it back to its original length (a mechanical deformation). The
thermal stress which develops if a cylindrical metal is completely constrained (not allowed moving
at all) is the product of the coefficient of linear expansion and the temperature change and Young’s
Modulus for material.
The heating of the test material is achieved by using an electrical resistance heater. The temperature
of the test specimen is measured by using a temperature sensor. The result of the temperature
reading is displayed on a digital temperature meter. The temperature meter doubles as a
temperature controller, allowing the desired testing temperature to be set.
This trainer is designed for easy installation and operation. Control panel is included in the unit.
All instrumentation and control components are mounted on the control panel.
36
Figure 1. LS-22056 Thermal stress apparatus
Specimen and Equipment
1. LS-22056 Thermal Stress Apparatus
2. Different test specimen
3. Dial gauge
Procedures
1. Switch ON the apparatus power supply (G).
2. Set the three dial gauges (C) reading to zero. Record these readings.
3. Record the initial temperature of each specimen by turning the selector (A) to T1, T2 and T3.
(T1 = stainless steel temperature, T2 = copper temperature, T3 = mild steel temperature).
4. Switching ON the heater (I). Set the temperature at the temperature controller (H) to about
60°C. Start the stop watch.
5. After 10 minutes elapsed, record down the three set temperature readings and the changed of
the length readings. (Note: Doesn’t matter if the temperature exceeds 60°C).
6. Repeat step 5 to obtain three set of results.
7. Compute the table given and calculate the thermal expansion coefficient for all the specimens.
Compare the readings with theoretical readings.
8. Calculate the test specimen strain value and hence the thermal stress value.
9. Plot the graph of thermal stress against temperature differences.
Useful information and equations
α = ∆L/(∆T)(L0)
Where,
α - Thermal expansion
∆L – Change of length
∆T – change of Temperature
L0 – initial length
Strain, ε = ∆L/L0
Thermal stress, σth = ε(E)
37
Results
1. Test specimen initial length, LO:
a. Mild steel = 263 mm
b. Copper = 263 mm
c. Stainless Steel = 262 mm
2. Calculate the thermal stress experience by the test specimen.
3. Plot the graph of thermal stress against temperature different for each specimen.
Table 1. Experiment result
38
Table 2. Difference of temperature and length results
Table 3. Coefficient of thermal expansion from experiment
39
Table 4. Thermal expansion comparison result
Table 5. Material’s strain
Table 6. Material’s Young’s modulus
Table 7. Material’s thermal stress respect to temperature difference
40
Discussion
1. Explain what is thermal expansion and thermal stress. Why this analysis is important when
selecting material.
2. From the graph obtained state down the finding.
3. Does the geometry (cross sectioned area) of a specimen affect the thermal expansion? Explain.
Conclusion
Give an overall conclusion based on the obtained experimental results.
41
B) BUCKLING OF STRUT
Introduction
In engineering, buckling is a failure mode characterized by a sudden failure of a structural member
subjected to high compressive stresses, where the actual compressive stress at the point of failure
is less than the ultimate compressive stresses that the material is capable of Withstanding. This
mode of failure is also described as failure due to elastic instability. Mathematical analysis of
buckling makes use of an axial load eccentricity that introduces a moment, which does not form
part of the primary forces to which the member is subjected.
This manual contains some fundamental theory for understanding the experiment, description of
the apparatus and experimental procedure to examine the supports reaction of the beam.
Objective
The objectives of this experiment are:
1. To determine the Buckling Load for a Pinned Ended Strut.
2. To determine the Buckling Load for a Fixed End Strut.
Theory
General
The ratio of the effective length of a column to the least radius of gyration of its cross section is
called the slenderness ratio (sometimes expressed with the Greek letter lambda, λ). This ratio
affords a means of classifying columns. All the following are approximate values used for
convenience.
• A short steel column is one whose slenderness ratio does not exceed 50; an intermediate
length steel column has a slenderness ratio ranging from 50 to 200, while a long steel column
may be assumed to have a slenderness ratio greater than 200.
• A short concrete column is one having a ratio of unsupported length to least dimension of
the cross section not greater than 10. If the ratio is greater than 10, it is a long column
(sometimes referred to as a slender column).
• Timber columns may be classified as short columns if the ratio of the length to least
dimension of the cross section is equal to or less than 10. The dividing line between
intermediate and long timber columns cannot be readily evaluated. One way of defining the
lower limit of long timber columns would be to set it as the smallest value of the ratio of
length to least cross-sectional area that would just exceed a certain constant K of the material.
Since K depends on the modulus of elasticity and the allowable compressive stress parallel
to the grain, it can be seen that this arbitrary limit would vary with the species of the timber.
The value of K is given in most structural handbooks.
If the load on a column is applied through the center of gravity of its cross section, it is called an
axial load. A load at any other point in the cross section is known as an eccentric load. A short
column under the action of an axial load will fail by direct compression before it buckles, but a
long column loaded in the same manner will fail by buckling (bending), the buckling effect being
so large that the effect of the direct load may be neglected. The intermediate length column will
fail by a combination of direct compressive stress and bending.
42
In 1757, mathematician Leonhard Euler derived a formula that gives the maximum axial load that
a long, slender, ideal column can carry without buckling. An ideal column is one that is perfectly
straight, homogeneous, and free from initial stress. The maximum load, sometimes called the
critical load, causes the column to be in a state of unstable equilibrium; that is, any increase in the
load, or the introduction of the slightest lateral force, will cause the column to fail by buckling.
The Euler formula for columns is:
F = π2EI / (KL)2
Where
F = maximum or critical force (vertical load on column),
E = modulus of elasticity,
I = area moment of inertia,
L= unsupported length of column,
K = column effective length factor, whose value depends on the conditions of end
support of the column, as follows.
For both ends pinned (hinged, free to rotate), K = 1.0.
For both ends fixed, K = 0.50.
For one end fixed and the other end pinned, K = 1.0/√2.0.
For one end fixed and the other end free to move laterally, K = 2.0.
(a)
(b)
Figure 1. (a) A column under a concentric axial load exhibiting the characteristic deformation of
buckling, and (b) Top platen – Pinned end condition
Description of TQR Strut Buckling Apparatus
The apparatus consists of a two columns frame with a fixed base and a movable top platen.
Specimen up to 1 meter length can be tested with either pinned or fixed end condition. A manual
screw jack is used for ease application of load to the specimen.
Specimen and Equipment
1. TQR Strut Buckling Apparatus
2. Vernier caliper
3. Measuring tape
4. Beams: Steel
43
Experiment 1: Determination of buckling load for a pinned ended strut
Theory
Figure 3. Bottom platen – Pinned end condition
Procedures
1. Switch on the digital indicator and warm it up for at least 10 minutes.
2. Choose a specimen and measure its length, width and thickness of the beam.
3. Calculate the theoretical buckling load for a strut with pinned end condition. This is to ensure
that the load applied to the strut does not exceed the buckling load.
4. Placed the grooved support into the slot of attachment for the end condition and tightened the
side screws. Refer to Fig. 2 for proper installation of the support.
5. Move the top platen upwards or downwards to bring the distance between the two supports
closer to the length of the strut.
6. Press the tare button on the digital indicator to set the reading to zero.
7. Place the specimen in the groove of the top support.
44
8. While holding the specimen, adjust the jack so that the end of the specimen just rest in the
groove of the bottom support.
(If the distance between the two supports is slightly less than the length of the strut, turn the
screw jack handle counter clockwise. If the distance between the two supports is slightly
greater than the length of the strut, turn the screw jack handle clockwise.)
9. Note the reading on the digital indicator. If the load is greater than 10N turn the jack handle
counter clockwise to bring it to less than 10N.
10. Check the position on the dial gauge to ensure that it is at the mid-length of the specimen. Set
the dial gauge reading to zero.
11. Press the tare button to set the load indicator to zero.
12. Load specimen in small increments by turning the screw jack handle slowly in the clockwise
direction.
13. For each load increments record the load and the corresponding mid-span deflection.
(Important: Please ensure that the applied load is always less than 80% of buckling load.)
14. Unload the specimen by turning the jack handle in the counter clockwise direction.
Results
1. Show all the measurements of beam.
i. Beam length L [mm]
ii. Beam width b [mm]
iii. Beam thickness h [mm]
iv. Moment of inertia of beam [mm4]
v. Theoretical Buckling Load Pcr [N]
2. Dial Gauge reading, 1 div = 0.01mm
3. From the data in Table 1 plot the graph of Deflection [mm] versus Deflection/load [mm/N].
4. Draw the best straight line through the points plotted.
5. From the plot determine the slope line. This is representing the buckling load for the specimen.
Table 1. Experimental results of buckling load for a pinned ended strut
Discussion
1. Assuming the value of E as 200GPa, calculate the theoretical critical buckling from the
following equation:
Pcr = π2El/L2
2. Discuss the result with the theoretical value.
Conclusion
Give an overall conclusion based on the obtained experimental results.
45
Experiment 2: Determination of buckling load for a fixed end strut
Procedures
1. Choose a specimen and measure its length, width and thickness.
2. Calculate the theoretical buckling load for a strut with fixed end condition.
3. Move the top platen upwards or downwards to bring the distance between the two supports
3. closer to the length of the strut.
4. Press the tare button on the digital indicator to set the reading to zero.
5. Placed the specimen in the slot of the upper attachment for end conditions. Refer to Fig. 1.
(If the distance between two attachments is less than the length of the strut, turn the screw jack
handle counter clockwise to lower the position of the attachment for the end condition. If the
distance is greater than the length of the strut, turn the screw jack handle clockwise to close
the gap.)
6. Same as procedure 9 to 14 in Experiment 1.
(a)
(b)
Figure 1. (a) Specimen positioning, (b) Assembly at the bottom platen-fixed end condition
46
Results
1. Show all the measurements of beam.
i. Beam length L [mm]
ii. Beam width b [mm]
iii. Beam thickness h [mm]
iv. Moment of inertia of beam [mm4]
v. Theoretical Buckling Load Pcr [N]
2. Dial Gauge reading, 1 div = 0.01mm
3. From the data in Table 1 plots the graph of Deflection [mm] versus Deflection/load [mm/N].
4. Draw the best straight line through the points plotted.
5. From the plot determine the slope line. This is representing the buckling load for the specimen.
Table 1. Experimental results of buckling load for a fixed end strut
Discussion
Compare and discuss the theoretical and experimental value.
Conclusion
Give an overall conclusion based on the obtained experimental results.
47
LAB 9: THIN CYLINDER
Introduction
This experiment gives students an opportunity to experiment with a cylinder that has a
diameter/thickness ratio of more than 10, making it thin-walled. The cylinder will undergo pressure
loading that will introduce hoop and longitudinal stresses on the surface of the material.
The fact that the cylinder is thin-walled allows for the assumption that the hoop and longitudinal
stresses are constant throughout the wall thickness or area. Two different conditions of pressure
loading will be tested: “open-end” and “close-end”. The open-end condition can be seen as
studying a portion of a long pipeline, while closed-end conditions can be imagined as looking at
an enclosed gas tank that holds a certain amount of pressure.
Using this computerized thin cylinder experiment we will introduce varying amounts of pressure
into the cylinder and utilizing strain gauge readings on the surface of the cylinder to determine
Young’s modulus (E), Poisson’s ratio (ν), and to study the strain Mohr’s circles of the two different
end conditions.
Objective
The objective of this experiment is to study the distribution of strains on the wall of a thin cylinder
with open and close end condition when subject to applied internal pressure.
Theory
Stress and strain in thin cylinder is calculated based on the membrane theory of thin cylindrical
shell.
Axial Equilibrium
Axial force at the end of the cylinder
Axial stress in the cylinder wall
Total axial force in the cylinder wall
For equilibrium
= pπr2
= σx
= 2πσxrt
= pπr2
σx = pr/2t
48
[1]
[2]
[3]
Circumferential Equilibrium
Internal pressure
= Circumferential stress
Consider an element of the shell making a small angle dθ
Radial force on this element is
= pπr2
Longitudinal stress, σx
= (εx + εy) (E/1-ν2)
= (70 x 109/0.91) (εx x 10-6 + εy)
Figure 1. Description of TQR thin cylinder apparatus
The apparatus consists of a thin aluminum having diameter of 75mm and thickness of 3.18mm.
The cylinder filled with oil and pressure is applied using hydraulic pump. A line valve is located
at the side of the pump and stop valve located in front of the cylinder. The stop valve is to stop the
oil from flowing back the cylinder while taking reading. By stopping the valve, the pressure can
be maintained at a constant value while reading is taken and must be open when the cylinder is
being pressurized and when the pressure is being released from the cylinder.
The line valve must be closed to pressure the cylinder and open to release the pressure from the
cylinder. The pressure is applied to the cylinder using the pump handle. A pressure gauge is to
indicate the magnitude of pressure applied to the cylinder.
Six strain gauges are provided to measure the resulting strain. Two gauges are to measure the strain
in circumferential direction (Hoop Strain). One gauge to measure the axial strain along the cylinder
and one gauge each to measure strain at approximately 30, 40 and 60 degrees. All gauges are
connected to six channels of the data logging system as Table 1.
The excitation voltage for the gauges is set at 1 volt to avoid heating of the gauges during the
experimental run.
A piston is available inside the cylinder to induce close and open end condition.
49
Figure 2. TQR thin cylinder
Table 1. Data logging channel
Specimen and Equipment
1. TQR Thin Cylinder Apparatus
2. A data logging system
Procedures
1. Switch on the data acquisition system.
2. Draw the piston out of the cylinder by turning the handle at the end of the cylinder to produce
an open-end condition.
3. Release all the pressure in the cylinder by opening the valve at the hand pump to allow the
hydraulic oil to flow from the cylinder to the pump reservoir. If the pressure gauge indicates
there is still pressure inside in the cylinder open the screw cap at the top of the pump reservoir
to allow trapped air escape.
4. When the pressure gauge reading in zero, close the screw cap and the line valve at the hand
pump.
5. Connect the wire from the apparatus to the data acquisition unit.
6. Check the communication between the computer and data acquisition, the 8000 Utility. Do not
change any parameter on the utility menu.
7. Start WinView CP32 software and change the sampling rate from 10 second to 2 second.
8. Click the Start button and click Overwrite button to record the initial strain readings of all the
gauges.
9. After 30 seconds click the Stop button. 15 readings will be recorded for each channel.
10. Close the Line Valve and open the Stop Valve.
50
11. Using the hand pump apply internal pressure to the cylinder until the pressure gauge indicates
15 bar.
12. Shut the Stop Valve to hold the pressure.
13. Click the Start button and click Append button to record the resulting strain reading due to
applied pressure.
14. Open the Stop Valve and increase the pressure in the cylinder.
15. Repeat step 10 to 14 until the pressure reached 35 bar.
16. Repeat another set of readings.
17. Open the Line Valve and valve at the hand pump to release the internal pressure at the end of
the experiment.
Results
1. Record the value of strain for Free End Condition and Fix End Condition in Table 2 and Table
3, respectively. Noted that all strain values in microstrain.
2. Calculation of stress from the strain reading:
a. Young’s Modulus = 70 GPa
b. Poisons Ratio = 0.3
c. Thickness of cylinder wall, t = 3.18 mm
d. Cylinder diameter = 75 mm
Table 2. Experimental results for free end condition
Table 3. Experimental results for fix end condition
51
3. Use Mohr Strain Circle method to obtain value of strain at 30°, 45° and 60°. Each angle and
pressure will require one circle to be drawn.
4. Plot the graph of strain εx against εy.
a. Draw the best fitting curve to pass through the plotted points.
b. From the plot determine the Poisson’s Ratio for the material.
5. Plot the graph εy against Pressure P.
a. Draw the best fitting curve through the plotted points.
b. From the plotted determine the slope and hence the E value for the cylinder material.
Discussion
1. Compare and discuss the theoretical and experimental value.
2. Define the relationship between the variables in this experiment.
3. Make comparison between the value of Poisson’s Ratio and Young’s Modulus obtain from the
experiment and that normally assumed in practice.
Conclusion
Give an overall conclusion based on the obtained experimental results.
52
LAB 10: THERMAL EXPANSION FOR SOLIDS & LIQUIDS
Introduction
An increase in temperature causes the vibrational amplitude of the atoms in the crystal lattice of
the solid to increase. The potential curve of the bonding forces corresponds only to a first
approximation to the parabola of the harmonic oscillation, generally it is flatter in the case of large
inter-atomic distances than in the case of small ones. If the vibrational amplitude is large, the centre
of the oscillation thus moves to larger inter-atomic distances.
The average spacing between the atoms increases, as well as the volume, V (at constant
pressure, p)
1 𝜕𝑉
𝛼 = ( ) 𝑝 (1)
𝑉 𝜕𝑇
Is called the volume of expansion coefficient; if we consider one dimension only, we obtain the
coefficient of linear expansion
1 𝜕𝑙
𝛼1 = ( )𝑝 (2)
𝑙 𝜕𝑇
Where is the total length of the body. Since the changes in length due to expansion,
∆𝑙 = 𝑙 − 𝑙𝑜
Are small compared with the original length, 𝑙𝑜 , we can say
∆𝑙 1
𝛼1 =
𝑙𝑜 Ɵ0
And thus,
𝑙 = 𝑙𝑜 [1 + 𝛼1 (Ɵ − Ɵ0 )]
Where Ɵ0 is the initial temperature.
The coefficients of linear expansion are tabulated in Table 1 below;
Type of Metal
Aluminum
Brass
Copper
Steel
Table 1. The coefficients of linear expansion of metals
𝛼1 /10−3 𝐾 −1
2.2
1.8
1.6
1.1
Objective
To determine the volume expansion of solids using the dilatometer.
Apparatus
1. Dilatometer
2. 2. Brass, iron and aluminum solid tube
53
z
Figure1: Dilatometer
Procedure
1. Connect the dilatometer as the experiment set up
2. Insert and clamp the solid tube by screw at fixing support. Set the scale on the dial gauge to
“0”
3. Fill in the water into the water bath and turn on the thermostat
4. Record the initial length, 𝑙𝑜 and its initial temperature, Ɵ0 for the solid tube. Increase the
temperature of the thermostat by 10 0C and Ɵ 𝑇 record the increased temperature,
5. Observe the gauge, if the value of the gauge changes. At this moment record the Ɵ 𝑇
length, . Determine the changes in length,
∆𝑙 = 𝑙 − 𝑙𝑜
6. Record the values for each increment of temperature until 80 0C
7. Repeat the steps for other solid tubes.
Results
Temperature,
C
K
10
20
30
40
50
60
70
80
0
Changes in length of the measuring tube, ∆𝒍 (mm)
Aluminum
Brass
Copper
Steel
1. Plot graph for relationship between length and temperature for aluminum, brass, copper and
steel.
2. Obtain the coefficient of linear expansion
from your experiment data.
Discussion and Conclusions
1. Compare coefficient of linear expansion from experiment data with that tabulated
in Table 1.
2. Discuss your findings.
54
LAB 11: HEAT CAPACITY (SOLIDS, LIQUIDS & AIR)
Experiment 1: Metals
Introduction
Under the assumption that no heat is released into nor gathered from the environment and also
that no other source of heat is active, the following is valid:
∆𝑄𝑚𝑒𝑡𝑎𝑙 = ∆𝐶𝑚𝑒𝑡𝑎𝑙 . 𝑚𝑚𝑒𝑡𝑎𝑙 . ∆Ɵ𝑚𝑒𝑡𝑎𝑙
∆𝑄𝑤𝑎𝑡𝑒𝑟 = ∆𝐶𝑤𝑎𝑡𝑒𝑟 𝑚𝑤𝑎𝑡𝑒𝑟 . ∆Ɵ𝑤𝑎𝑡𝑒𝑟
The metal sample gives an amount of heat up in the calorimeter and the water takes on the amount
of heat. The two amounts must be equal. The specific heat capacity is hereby the only unknown,
the other quantities were either measured or are assumed to be known. We therefore have:
∆𝑄𝑚𝑒𝑡𝑎𝑙 = ∆𝑄𝑚𝑒𝑡𝑎𝑙
𝐶𝑚𝑒𝑡𝑎𝑙 . 𝑚𝑚𝑒𝑡𝑎𝑙 . . ∆Ɵ𝑚𝑒𝑡𝑎𝑙 = 𝐶𝑤𝑎𝑡𝑒𝑟 . 𝑚𝑤𝑎𝑡𝑒𝑟 ∆Ɵ𝑤𝑎𝑡𝑒𝑟
𝐶𝑚𝑒𝑡𝑎𝑙 = ∆𝐶𝑤𝑎𝑡𝑒𝑟 .
𝑚𝑤𝑎𝑡𝑒𝑟
.
∆Ɵ𝑤𝑎𝑡𝑒𝑟
𝑚𝑚𝑒𝑡𝑎𝑙 ∆Ɵ𝑚𝑒𝑡𝑎𝑙
(1)
(2)
When the values obtained in the experiment are entered, we have the following:
𝐶𝑚𝑒𝑡𝑎𝑙 =
4.2𝐽 150𝑔 ∆Ɵ𝑤𝑎𝑡𝑒𝑟
0ᵒ𝐶𝑔
.
60𝑔
.
∆Ɵ𝑚𝑒𝑡𝑎𝑙
Table 1: Literature values
Objectives
To determine the specific heat for various metals.
55
(3)
Apparatus
1) Cobra4 wireless manager
2) Cobra4 wireless link
3) Cobra4 sensor unit temperature
4) Support rod
5) Universal clamp
6) Metal bodies (aluminum, steel, brass)
7) Agitator rod
8) Felt sheet
9) Erlenmeyer flask
10) Beaker
11) Burner
12) Boiling chips
13) Cobra4 software
Procedures
1) Set up the stand up as in Fig.1
2) Fill at least 250ml of water in Erlenmeyer flask and add two boiling chips
3) Thread a fishing line through each of the three metal bodies and knot it to make a loop
4) Hang all the metal bodies using universal clamp and soak the metal bodies in the water inside
the Erlenmayer flask.
5) Make up a thermally insulating vessel (calorimeter) using two glass beakers and two felt
sheets.
6) Fill 150ml of water in the calorimeter.
56
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
Bring the water in the Erlenmeyer flask to boil, adjust the flame so small that it just continues
to boil.
Start the PC and the window operating systems
Plug the Cobra4 wireless manager to the USB interface of the PC
Plug the Cobra4 sensor unit temperature to the Cobra4 wireless link
Switch on the wireless link, the sensor is logged on in the “Navigator”
Insert the temperature sensor through a hole in the lid of the calorimeter so that it dips into
the water but does not touch the bottom
Start measurement recording in measure “•“, a measured temperature value is now recorded
every second
Stir and wait until the temperature display remains constant, at least 100s
Take the metal body and rapidly transfer it to the calorimeter, immediately put back the lid
Carefully stir the water in the calorimeter so that the water heat is uniformly distributed
End measurement with “ ”, when the temperature remains constant or when it slowly
decreases.
Use the “Display option” button to enter the suitable name for the measurement series under
“Title”
Pour away the water in the calorimeter, dry the beaker and fill 150ml of water to it.
Repeat experiment in the same way with each of the two metal bodies
Finally measure the temperature of the boiling water before you turn off, holding the
temperature sensor in the universal clamp.
Results
1) Select “survey” tool in the main measure programme to determine the initial temperature in
the calorimeter and the mixture temperature in the calorimeter for each of the three
measurement curves.
2)
Also enter the temperature of the boiling water, which depends on the athmospheric pressure
and the salt content of the water. Calculate the temperature differences between the
temperature sample and the temperature of the water in the calorimeter before and after
transferring the metal sample into the calorimeter.
57
Table 2: Measurement Record
3)
Calculate the heat capacities of the metal samples using equation (3).
Discussion and Conclusions
1)
2)
3)
4)
Compare your results with the literature values given in Table 1. Why does equation (3)
lead to values that are too low?
Which heat capacity was not take into consideration?
Where are the greatest measurement uncertainties?
Conclude your findings.
58
Experiment 2: Liquids
Introduction
At constant heating voltage U, the temperature T increases linearly with the heating time t. the
power P of the heating coil is calculated from the measured values of the heating voltage U and
the heating current I from P=U.I and is in this case almost constant. The added energy Q, with
Q=P.t , heats the water of mass m and specific heat capacity c by the temperature T, whereby the
following is valid:
𝐶=
𝑄
∆𝑇
(1)
The specific heat capacity of amaterial on per mass basis
𝐶=
𝜕𝐶
(2)
𝜕𝑚
Which in the absence of ohase transitions is equivalent to
𝐶
𝐶 = 𝐸𝑚 =
(3)
𝑚
𝑄 = 𝐶𝑚∆𝑇 (4)
Q is therefore the calculated amount of heat over the time t, that was given up to the liquid in the
calorimeter. Both T and Q run linearly over time t, so that according to equation (1), a constant
value must be given for c.
59
Increase in the differential temperature T and the added heat quality Q over the time t.
The determination of the specific heat capacity c is correspondingly carried out from the
temperature increase (ΔT) and the amount of heat added (ΔQ) with equation 1.
Objective
To determine the specific heat capacity of water.
Apparatus
a.
Software Cobra4
b.
Cobra4 wireless link
c.
Cobra4 sensor unit manager
d.
Multi-range meter
e.
Heating coil with sockets
f.
Glass beaker 400ml
g.
Erlenmeyer flask 250ml
h.
Glass beaker 250ml
i.
Pipette with rubber bulb
j.
Felt sheet
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Figure 1: Specific heat capacity of water apparatus
Procedure
1)
2)
3)
4)
5)
Set up the experiment as shown in the photo at Fig. 1 and the circuit diagram at Fig. 2.
Fill 200g of distilled water at room temperature into the calorimeter and add the magnetic
stirring bar. Put the lid on the calorimeter. Lead the temperature sensor through the opening
in the lid. The switch must be open to start with.
Start the PC and windows.
Plug the Cobra4 wireless link. The two sensors are automatically recognized and are assigned
an ID number that can be seen in the display on that particular wireless link.
Start the measure software package on PC.
61
6)
Load experiment (Experiment>Open experiment>2.physics). All necessary settings for direct
recording of measured value are now loaded. The difference between the tared starting
temperature and the current temperature is then displayed (Fig. 3).
7) The electric power P is also calculated as the product of the applied voltage U and the resulting
current I as well as the amount added heat Q from Q=P.t.
8) Start measured value recording in measure • (Fig.4).
9) Switch the magnetic stirrer on.
10) Adjust the heating voltage U to approximate 10 V and close the switch. Measure time
approximate 5 minutes.
11) End measurement and transmit the measured values to the “measure” main programme for
further analysis.
62
63
Results
1) Plot graph for temperature T versus time t, and energy Q versus time t.
2) Analyze your from the experiment.
3) Compare your with theoretical c value. Discuss your analysis.
Discussion and Conclusions
1) Is the experimental
obtained within the range? Why?
2) Can the experimental results be accepted or rejected? Why?
3) Conclude your findings.
64
Experiment 3: Gas
Introduction
The first law of thermodynamics can be illustrated particularly well with an ideal gas. This law
describes the relationship between the change in internal energy the heat exchanged with the
surroundings and the work performed by the system generally speaking. In our case the work
being performed is the pressure-volume work results into an volume increase keeping constant the
pressure,
.
()
The molar heat capacity
of a substance results from the amount of absorbed heat
temperature change per mole where
is the number of moles:
()
and the
One distinguishes between the molar heat capacity at constant . According to equation (1) and (2)
and under isochoric conditions ), the following holds true:
()
It is obvious that from equation (3) that the molar heat capacity is a function of the internal energy
of the gas. The internal energy can be calculated with the aid of the kinetic gas theory with the
number of degrees of freedom and the universal gas constant :
()
Taking equation (3) into consideration it follows that:
()
Objective
To determine the molar heat capacity of air at constant volume.
65
APPARATUS
1)
2)
3)
4)
5)
6)
7)
8)
Manometer
Digital multimeter
Aspirator bottle 10000 ml
Stopcock 1-way
Stopcock 3-way
Nickel electrode
Chrome nickel wire
Two-way switch
Procedures
1) Connect the manometer to the bottle with piece of tube
2) Ensure the 3-way stopcock inserted on top of the glass and only 1-way will be use
3) Set up the connecting as in Figure 1. Set the digital counter meter
4) Connect the ammeter and the voltmeter by series and parallel respectively
5) The voltmeter is connected in parallel with the nickel electrode. Meanwhile the
ammeter is connected with the nickel electrode and 2-way switch
6) Ensure that the stopcock at the bottom of the glass is closed
7) Start the measurement by activating the push start button switch on the universal counter
and the switch the two-way switch to Start.Stop (Gate1)
8) At this moment, the pressure bar on manometer will increase. When the pressure is
increase to 0.1mbar, switch the two-way switch to Stop (Gate 2). Record the time.
9) Equalize the pressure by opening the 3-way stopcock. Push the zero button on the
universal counter.
10) Repeat step 7 and 8 for 0.2 mbar, 0.3 mbar and so on.
66
Results
1)
Tabulate the increment in pressure with time in a table.
2)
Plot a graph for pressure versus time.
3)
Analyze your data.
4)
From your data experiment, determine .
Discussion and Conclusions
1)
Is the experimental obtained within the range? Why?
2)
Can the experimental results be accepted or rejected? Why?
3)
Conclude your findings.
67
LAB 12: TEMPERATURE MEASUREMENT
Introduction
There are numerous ways of measuring temperature. These utilise various physical processes to
acquire temperatures. The indication of the value measured can be direct. In this case the
temperature acquisition medium is linked to a scale. In industry, temperatures are often measured
by electronic means. The measured values are indicated on digital displays. In addition, the values
measured are converted into standardised electrical signals so that the temperature information can
be supplied to remote displays and other equipment for further processing (controllers).
Different types of thermometers and electronic sensors are contained in the WL 202 Fundamentals
of Temperature Measurement unit. The values measured by the sensors can be read off on digital
displays. At the same time, a standardised signal (0...10 V) is output on laboratory sockets so that
the change in the measured values over time can be recorded using a recorder. The unit includes
several heat sources that also enable higher temperatures to be generated. In this way, it is ensured
that the measuring range in which the individual temperature measuring devices are normally used
can be reached.
Objective
To measure temperature using various type of temperature sensors.
Apparatus
68
Procedure
1. Turn on the WL 202 at the master switch.
2. Fill the insulated tank with ice and a small amount of ice water.
3. Turn on the relevant switch above the socket. Connect the laboratory heater at the socket. Place
a container of water on the laboratory heater. Bring the water to the boil and record the
measured values using various types of thermometer sensors.
4. Record your value.
Results
69
Discussion and Conclusions
1. Compare your measurement using all temperature sensors.
2. From your opinion which sensor is accurate? Explain.
70