Eastern Mediterranean University
Department of Mechanical Engineering
Laboratory Handout
COURSE: Dynamics ME 233
Semester: Spring (2007-2008)
Name of Experiment: Investigating impulse and momentum for rigid bodies
Instructor: Assoc. Prof. Dr. Fuat Egelioglu
Assistant: EHSAN KIANI
Take Home Laboratory Project
Submitted by:
Student No:
Group No:
Date of experiment:
Date of submission:
------------------------------------------------------------------------------------------------------
EVALUATION
Activity During Experiment & Procedure
30 %
Data , Results & Graphs
35 %
Discussion, Conclusion & Answer to Questions
30 %
Neat and tidy report writing
5%
Overall Mark
Name of evaluator: EHSAN KIANI
Objective
To be able to demonstrate an understanding of:



Interactions from the perspective of impulse and momentum
That impulse causes a change in momentum (just like force causes a change in
velocity)
The effect of applying a force over a period of time – Impulse
Activity:
1. Rank the objects (A – E) in order of increasing momentum.
mA  30 g
vA  3 m / s
mC  300 g
mB  40 g
mD  10 g
vB  5 m / s
Least 1________
vC  2 m / s
2________
3________
2
vD  5 m / s
4________
mE  20 g
vE  7 m / s
5________ Greatest
2. The Position vs. Time graph is shown for an object of mass 0.60 kg. Draw the
corresponding Momentum vs. Time graph. (Use an appropriate numerical scale for your
Momentum vs. Time graph).
p (kg  m / s )
x ( m)
20
15
10
t ( s)
5
2
2
4
6
8
4
6
8
t ( s)
3. The Momentum vs. Time graph is shown for an object of mass 500 g. Draw the
corresponding Acceleration vs. Time graph. (Use an appropriate numerical scale for your
Acceleration vs. Time graph).
p (kg  m / s )
a(m / s 2 )
20
15
10
t ( s)
5
2
2
4
6
8
t ( s)
3
4
6
8
Below are four graphs of Force vs. Time for a 1.0 kg object that moves to the right with a
speed of 1.0 m/s. What are the velocities (speed and direction) after the impulse?
4.
5.
F (N )
F (N )
4.0
4.0
2.0
2.0
t ( s)
t ( s)
1.0 s
2.0 s
6.
7.
F (N )
F (N )
4.0
2.0
0.50 s
2.0 s
t ( s)
t ( s)
2.0 s
2.0
4.0
4
Questions 7-8 refer to Figure 1 below which depicts a pendulum that swings down from
some height and collides with an aluminum post (the pendulum bob is moving
horizontally when it strikes the post). The post is not secured to anything and is free to
fall if hit hard enough by the pendulum. The point is to knock the post over, so you’re
given a choice of two types of material for your pendulum; one is a bouncy rubber ball,
and the other is a wad of sticky clay. Both have the same mass m, and you may assume
that both hit the post with the same speed (we will cover this point once we get to
energy).
10. Which will you choose if you want to
knock the post over? Why?
Figure 1.
11. Both balls hit the post with the same initial momentum pix but one bounces (the
rubber ball) with no loss of speed and the other (the clay ball) sticks. What are the final
momenta of each?
Rubber Ball: p fx ,rubber = _________
Clay Ball: p fx ,clay = _________
12. Now find the change in momentum for each ball.
Rubber Ball: px ,rubber = _________
Clay Ball: p x ,clay = _________
5
13. Which ball experiences the greater (in magnitude) impulse? Explain.
14. Which law of physics will tell us how the post is affected? In other words, which
ball exerts a larger impulse on the post?
15. How does this affect your answer to question #10? If you’ve changed your mind
about the question, then tell why?
6
Error Analysis
Since the aim of this experiment is to get students more familiar with the mechanical
concepts of “Impulse and Momentum” in a theoretic basis, error analysis should be made
in an individual way of conceiving error sources and degree of accuracy.
As for each measurement, variety of factors are involved it would be better to increase
the measurement time in order to enhance the preciseness.
7
Eastern Mediterranean University
Department of Mechanical Engineering
Laboratory Handout
COURSE: Dynamics ME 233
Semester: Fall (2008-2009)
Name of Experiment: Measurement of Elastic Constant of Spiral Spring, and
Earth’s Gravitational Intensity
Instructor: Assoc. Prof. Dr. Fuat Egelioglu
Assistant: EHSAN KIANI
Submitted by:
Student No:
Group No:
Date of experiment:
Date of submission:
------------------------------------------------------------------------------------------------------------
EVALUATION
Activity During Experiment & Procedure
30 %
Data , Results & Graphs
35 %
Discussion, Conclusion & Answer to Questions
30 %
Neat and tidy report writing
5%
Overall Mark
Name of evaluator: EHSAN KIANI
8
1.
OBJECTIVES
The aim of the experiment is to find kinetic elasticity of spiral springs and Earth’s
Gravitational Intensity.
2.
APPARATUS
Light spiral spring A, scale-pan B, meter rule C, two clamps and stands, two boxes of
weights, light pointer P spring, stop-watch.
3.
THEORY
So far we have concentrated on kinematics, the description of motion. For this
exercise we will examine some of the causes of motion, a subject called dynamics.
Newtonian dynamics centers around forces, which produce accelerations. Forces, such as
gravity, rubber bands or friction, are part of the environment of the object we are
studying. When one or more forces act on a mass it accelerates according to the
celebrated Newtonian law There are several aspects of this law that can be studied. A
force of constant magnitude evidently produces a constant acceleration. Also, for
constant force the acceleration should be inversely proportional to the mass of the
accelerated object. Perhaps odder, it seems that the acceleration in some chosen direction
is proportional to the vector component of the force in that direction, independent of
other components. All of these effects can be observed without instruments, as well as
being quantitatively measurable.
9
4.
WORK TO BE CARRIED OUT
Suspend the light spring from the clamp of the stand and attach a light pointer P to the
spring. Set up a fixed vertical meter rule C beside the spring A, attach B to the spring,
and then add suitable weights, noting the reading of the pointer each time. Do this for
about eight loads on the scale-pan. Then remove each weight, and record the reading of P
each time. If the spring has not been permanently strained the reading of P will return to
its original or zero reading when all the weights and the scale-pan have been removed.
Weight the scale-pan.
5.
EXPERIMENTAL DATA
Mass on
scale-pan/Kg
Reading on
meter rule/mm
Total mass/Kg
10
Extension/mm
6.
DATA ANALYSIS
Add the scale-pan mass to each load to find the total mass and enter the results in the
table. Use two decimal accuracy for your data, if it is needed.
7.
GRAPH
Plot the extension in meter v. the mass T whose weight extended the spring draw the best
straight line through the origin. You can use related softwares inorder to draw the
diagrams. From the graph calculate the mass  per meter; this is b/a.
8.
DISCUSSION, CONCLUSION & ANSWER TO QUESTIONS
9. Error Analysis
The knowledge we have of the physical world is obtained by doing experiments and
making measurements. It is important to understand how to express such data and how to
analyze and draw meaningful conclusions from it.
In doing this it is crucial to understand that all measurements of physical quantities are
subject to uncertainties. It is never possible to measure anything exactly. It is good, of
course, to make the error as small as possible but it is always there. And in order to draw
valid conclusions the error must be indicated and dealt with properly.
If the result of a measurement is to have meaning it cannot consist of the measured value
alone. An indication of how accurate the result is must be included also. Indeed, typically
more effort is required to determine the error or uncertainty in a measurement than to
perform the measurement itself. Thus, the result of any physical measurement has two
essential components: (1) A numerical value (in a specified system of units) giving the
best estimate possible of the quantity measured, and (2) the degree of uncertainty
associated with this estimated value. For example, a measurement of the width of a table
would yield a result such as 95.3 +/- 0.1 cm.
F1=mxg=(0.3)kgx(9.81)m/s2 = 2.943 kg.m/s2=2.943 N
F1=mxg=(0.5)kgx(9.81)m/s2 = 4.905 kg.m/s2=4.905 N
11
F1=mxg=(0.8)kgx(9.81)m/s2 = 7.848 kg.m/s2= 7.848 N
F1=mxg=(1.3)kgx(9.81)m/s2 = 12.753 kg.m/s2=12.753 N
F1=mxg=(1.8)kgx(9.81)m/s2 = 17.658 kg.m/s2=17.658 N
F1=mxg=(2.1)kgx(9.81)m/s2 = 20.601 kg.m/s2=20.601 N
F1=mxg=(2.8)kgx(9.81)m/s2 = 27.468 kg.m/s2=27.468 N
F1=mxg=(3.3)kgx(9.81)m/s2 = 32.373kg.m/s2=32.373 N
∑ % Error  100% 
Actual  Experiment
Actual
Maximum elongation is
E(s) = x±y ≈ 33±0.001 = 32.999

% Error 
33.001
 1.00003030
33
Error: 0.6875x 100 = %68.75
12
Eastern Mediterranean University
Department of Mechanical Engineering
Laboratory Handout
COURSE: Dynamics ME 233
Semester: Spring (2006-2007)
Name of Experiment: Measurement of the Coefficients of Static and
Dynamic Friction
Instructor: Assoc. Prof. Dr. Fuat Egelioglu
Assistant: EHSAN KIANI
Submitted by:
Student No:
Group No:
Date of experiment:
Date of submission:
------------------------------------------------------------------------------------------------------------
EVALUATION
Activity During Experiment & Procedure
30 %
Data , Results & Graphs
35 %
Discussion, Conclusion & Answer to Questions
30 %
Neat and tidy report writing
5%
Overall Mark
Name of evaluator: EHSAN KIANI
13
10. OBJECTIVES
The aim of the experiment is to measure coefficient of static and dynamic friction and to
investigate the laws that govern friction.
11.
APPARATUS
Wooden block A with a hook attached, a plane piece of wood B with a grooved wheel C
at one end, scale-pan S, light string, weights, boxes of weights, spring balance.
Fig. 1
12.
THEORY
We encounter friction at almost all times during the day. Friction between our foot and
the floor helps us walk. In spite of its importance, friction is still not well understood. However,
empirical laws describe the friction between two surfaces. These laws are as follows:
1.The ratio of the maximum frictional force and the normal force is a constant and equals the
coefficient of friction, μ, and depends only on the nature of the two surfaces in contact.
I.e.: μ(Frictional Force) / (Normal Force).
14
Fig. 2
2.The coefficient of friction is independent of the area of contact.
3.The coefficient of kinetic friction μ k (the object is in motion) is lower than the coefficient of
static friction μ s(the object is stationary.)
We will first use the configuration shown in Fig. 2 to determine the coefficient of static and
kinetic friction between a few surfaces. Here, the normal force N = Mg, obtained by balancing
forces in the vertical direction on the block. Recall that the pulley only changes the direction of
force but does not change its magnitude. Balancing forces in the horizontal direction, we
obtain:
mg – μ N = 0.
Therefore,
μ = m/M.
Next, we explore if there is a substantial change in  if the surface on which the block is
sliding is at an angle to the horizontal. In this case the normal force N is not equal to Mg,
but rather to Mg cos. Balancing forces along the inclined plane when the block is about to
move up the plane, we obtain:
mg - N – Mg sin = 0 .
Substituting for N, we obtain:
 = (m/M – sin)/cos .
(Note: When the block is about to move downwards, the direction of the frictional force
is in opposite direction and therefore you will have to modify the formula appropriately.)
Fig. 2
Kinetic Friction:
6. Next, we determine the coefficient of kinetic friction (you may use either the
wooden or felt side.) The procedure is the same as before, except that after adding an
15
incremental mass to the hanger, give a gentle push to the block. If the block moves away
with a constant speed, then the tension in the string corresponds to the kinetic frictional
force. Note, you should take care not to add too much mass in which case the block will
accelerate to the right and you will erroneously easure a higher kinetic frictional force.
How does k compare with s.
7. Using the smaller of the two surfaces, determine the k (and time permitting
static friction) how does it compare with k using the larger surface?
Friction on an inclined plane:
8. Tilt the Aluminum track through approximately 30 o (you may use the angle
indicator to approximately set this angle but, measure the height and length of an
appropriate angle to determine the angle more accurately.) You may use the larger side of
the wooden block for these measurements. (Does the block move up or down the slope
with just the hanger in place? If the block moves up the incline, chose a larger angle of
inclination such that the block moves downwards.) Add masses in small increments to the
hanger so that the block stops sliding down. Which direction is frictional force acting?
Determine the coefficient of friction.
9. (Optional) Next, add small increments of masses on the hanger, the block
will be stationary, and then at a critical mass m, the block will move up the slope. Does the
frictional force change as you add masses? Which direction is the frictional force pointing?
Determine the coefficient of friction and compare to the values you obtained earlier for the
same surfaces. What can you conclude from your experiments regarding the nature of the
coefficient of friction and the its dependence on the type of materials and conditions.
Questions:
1. In which direction does the frictional force act under your foot as you are
walking forward?
2. Can the coefficient of friction be greater than 1.0?
3. How does your measurement of static and coefficient of friction explain the
superiority of anti-lock brakes (as opposed to regular brakes?)
16
13.
WORK TO BE CARRIED OUT
Weight the block A and the scale-pan S on the spring balance. Attach the scale-
pan to the hook of A by light string passing round the wheel C. Mark the position of A on
board B with pencil. Then gently add increasing weighs to S until A, and by adding
increasing weights to S, again record the total weight in S when A begins to slip. Repeat
for two more increasing weights on A, returning the block A to its original place on B
each time.
In Dynamic friction with the apparatus shown in the Fig.1 place a weight on S and give
A a slight push towards C. Add increase weight to S, giving A a slight push each time. At
some stage, A some stage A, will be found to continue moving with a steady, small
velocity. Record the corresponding weight in the scale-pan S. Now increase the reaction
of B by adding weight to A, and repeat for two more weights on A, returning the block to
its original place on B each time.
14. EXPERIMENTAL DATA
Normal
reaction R/gf
Weight in scale-pan
on slipping /gf
Limiting
frictional
force, F/ gf
17
Dynamic
Normal
reaction
R/gf
Weight in
scale-pan
on moving
A /gf
Friction
force, F’
/gf
15.
DATA ANALYSIS
Fill out third to sixth column of the above table.
16.
GRAPH
Plot F’ v. R.
The gradient, a/b=  ’=….
17.
DISCUSSION AND CONCLUSION
18.
Error Analysis
In numerical simulation or modeling of real systems, error analysis is concerned with the
changes in the output of the model as the parameters to the model vary about a mean.
For instance, in a system modeled as a function of two variables z = f(x,y). Error analysis
deals with the propagation of the numerical errors in x and y (around mean values and
) to error in z (around a mean )
∑ % Error  100% 
Actual  Experiment
Actual
Types of Error
Systematic: Defective equipment or improper measurement technique can
shift all measurements from best known values. If all your measurements
are much lower than that expected, for example, there may be a systematic
error.
Random: Errors that fluctuate from one measurement to the next. Some
sources are instrument sensitivity, noise, and statistical processes. This error
shifts the results in an arbitrary direction.
Mean Value
18
A practical guide in expressing errors that are randomly distributed is to
state errors according to the number of repeated measurements done. If the
measured quantity is X, and the average is Xave, the result may be expressed
as:
Number of Measurements (N)
Result
X  Sample Deviation
1
2-10
Xave  Average Deviation
>10
Xave  Standard Deviation
Xave  Standard Deviation of
the Mean
where
Sample Deviation = X
Average Deviation =
Standard Deviation =
1
N
| X
X |
 ( Xave  X )
Standard Deviation of Mean =
19
ave
2
N 1
 ( Xave  X )
N ( N  1)
2
20