UNESCO-NIGERIA TECHNICAL &
VOCATIONAL EDUCATION
REVITALISATION PROJECT-PHASE II
NATIONAL DIPLOMA IN
MECHANICAL ENGINEERING TECHNOLOGY
MECHANICAL ENGINEERING
SCIENCE[STATICS]
YEAR I- SE MESTER I
PRACTICALS
Version 1: December 2008
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PROGRAMME: MECHANICAL ENGINEERING SCIENCE (STATICS-MEC 111)
COURSE SPECIFICATION PRACTICAL CONTENT
TABLE OF CONTENTS
Week 1
1. Experiment No 1: Parallelogram of Forces
Week 2
2.
Experiment No 2: Triangle of Forces
3.
Experiment No 3: Polygon of Forces
Week 3
Week 4
4. Experiment No.4: Calculation of coefficient of Friction
Week 5
5. Experiment No.5: Friction on an Inclined Plane
Week 6
6. Experiment No.6: Angle of Friction
Week 7
7. Experiment No.7: Sliding Friction
Week 8
8. Experiment No 8: Principle of Moments
Week 9
9. Experiment No 9: The Pivot (or beam) balance
Week10
10. Experiment No.10 Forces in Frame Structures
Week11
11. Experiment No. 11 Finding the point where an Object’s mass acts
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Week12
12. Experiment No.12 mo r e s c i e n c e o f b a l a n c e
Week13
13. Experiment No. 13 to verify the laws of limiting friction
Week14
14. Experiment no.14 to verify Lami’s theorem
Week15
15 Experiment no.15 to verify triangle law ii
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EXPERIMENT NO 1: PARALLELOGRAM OF FORCES
Teaching Element:
When two forces act on a body in different directions in one plane, they are equivalent to
single force (the resultant) acting somewhere in between them. An example of this is
when a sledge is pulled by two horizontal ropes spread at an angle. The sledge will move
in a direction between the ropes along the line of their resultant force. Until the sledge
moves, it will pull back against the ropes with a single horizontal force equal and
opposite to the resultant of the two rope forces.
It can be shown that when three such forces are balanced (that is in equilibrium), their
lines of action all meet at a point. Using this fact, the resultant of two forces in the same
plane at an angle can be found by a graphical method called the Parallelogram of Forces.
Student Objectives:
The object of this experiment is to test that when three non-parallel forces in the same
plane are in equilibrium, their lines of action meet at a point, and hence to show that the
resultant of two forces can be found using the Parallelogram of Forces.
Apparatus:
Force board, weights, metal ring, cord, pulleys, drawing-paper, and drawing-pins
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Parallelograms of Forces
Method
1. Pin a sheet of drawing-paper to the board.
2. Fix the pulleys in any position and suspend weights so that the cords are at rest.
3. Note the values of the three weights.
4. Make a mark at the centre of the ring and one under each cord. (Care must be
taken to ensure that the eye is placed level with, and directly in front of, the point
at which the mark is being made.)
5. Remove the drawing-sheet.
6. Join the central mark (O) to each of the other three by straight lines. Put an arrowhead on each line to show the sense of the force, and indicate beside the line the
weight at the end of the cord.
7. Choose a suitable scale, and mark a length along OA to represent the force which
acted on the corresponding cord. Repeat for line OB.
8. Complete the parallelogram AOBC, and join OC.
9. Using the chosen scale, find the force represented by OC. This is the resultant of
the forces in the cords OA and OB.
Conclusion
1. What is the magnitude of the resultant found by this method?
2. What is the direction of the resultant?
3. Are the resultant and the central weight equal in magnitude and opposite in
direction?
4. State the theorem on which the method is based.
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EXPERIMENT NO 2: TRIANGLE OF FORCES
Teaching Element:
When three forces in the same plane act in different directions on a stationary body their
lines of action meet at a point. Because of this the forces can be represented by a force
diagram called the Triangle of Forces. This can be used to find the size of two of the
forces when the third force is known.
Student Objectives:
The object of this experiment is to test that three non-parallel forces in equilibrium can be
represented by a triangle of forces, from which two of the forces can be found when the
third force is known, provided that the direction or line of action of the forces is known.
Apparatus: as in Experiment No.1
Triangle of Forces
Method
1. Pin a sheet of drawing-paper to the board.
2. Fix the pulleys in any position and suspend weights so that the cords are at rest.
3. Note the values of the three weights.
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4. Make a mark at the centre of the ring and one under each cord. (Care must be
taken to ensure that the eye is placed level with, and directly in front of, the point
at which the mark is being made.)
5. Remove the drawing-sheet.
6. Join the central mark (A) to each of the other three by straight lines. Put an arrowhead on each line to show the sense of the force, and indicate beside the line the
weight at the end of the cord.
7. Produce, beyond A, the line representing Q, and on it mark a length AC to
represent the force Q to scale.
8. Along the cord AD mark a length AB to represent the force P to scale.
9. Join BC.
Conclusions
1.
2.
3.
4.
5.
Is BC parallel to the cord carrying load R?
Does BC represent the force R drawn to scale?
Can the forces be represented by the sides of a triangle?
What do you notice about the arrow-heads on the forces in the triangle ABC?
State the theorem known as the “Triangle of Forces.”
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EXPERIMENT NO 3: POLYGON OF FORCES
Teaching Element:
In the design of pin-jointed plane structures such as girders, bridges and roof trusses, it is
necessary to find the forces acting in each member so that the frame can be made strong
enough to withstand the maximum loads exerted upon it. The Polygon of Forces is
frequently employed to find such forces and deals with each joint in turn. This
experiment could be regarded as ONE such joint on a structure, and it will be shown that
in a system containing four or more forces, two unknowns can be found in magnitude or
direction if the remaining information is known. The Polygon of forces is an extension of
the Triangle of Forces, and whereas Tri means three, Poly means many.
Student Objectives:
The object of this experiment is to test that when four or more forces are in equilibrium at
a point, they can be represented by a Polygon of Forces from which unknown forces can
be found.
Apparatus:
Force board, several weights, metal ring, cord, pulleys, drawing-paper, and drawing-pins.
Polygon of Forces
Method
1. Pin a sheet of drawing-paper to the board.
2. Fix three pulleys in any position on the board and hang weights on the cords as
shown in Fig.19, so that the forces P, Q, R and S are in equilibrium.
3. Adjust the cords slightly if necessary, so that each is in line with the centre of the
ring.
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4. Make a mark on the paper at the centre of the ring, and one directly below each
cord. Note the values of the forces.
5. Remove the paper and join the central mark to the other marks by straight lines.
Put an arrow-head on each line the sense of the force and indicate beside each line
the weight at the end of that cord.
6. Represent the forces Q, S and R in order (i.e. the arrows following each other) as
shown in Fig. 20.
7. Join the free ends of the lines representing Q and R, and measure the line to
scales.
Conclusions
1. How does the measured line compare with the observed value of P, in magnitude,
sense and direction?
2. State the theorem you set out to illustrate.
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EXPERIMENT NO. 4 FRICTION - CALCULATION OF
COEFFICIENT OF FRICTION
Objective:
Determination of the coefficient of kinetic friction for pine on hardwood, and
for leather on hardwood.
Apparatus:
• Inclined hardwood plane,
• Various wood blocks,
• Blocks with leather on the bottom,
• Blocks with sand paper on the bottom,
• Standard weights, and
• Balance.
Theory:
The smoothest solid surfaces are still uneven (microscopically or otherwise).
Therefore, it takes some force to move bodies across each other when they are in contact,
either because particles are being broken off (wear), or because the bodies are being
separated slightly because of the projections on the surfaces. This is friction.
Frictional resistance between surfaces is:
1. proportional to the force pressing the bodies together,
2. dependent on the nature of the surface,
3. independent of contact area, and
4. Independent (within limits) of the relative speed between the surfaces.
Hence, for any given surface, the coefficient of kinetic friction is:
where:
• Ff - is the frictional force resisting motion,
• N - is the normal force perpendicular to the surface,
• mw - is the mass of the weight in kg attached on the string, and
• m - is the mass of the wooden block and mass on top of it.
Force Ff is parallel to the surfaces in contact and the normal force N is
perpendicular to the surfaces in contact. Thus, friction force Ff is
perpendicular to normal force N.
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Procedure:
1. Weigh the blocks that you will use to slide down the board and record the
masses on the data table.
2. With the plane horizontal and secured, attach a weight (mw) to a block
lying on the board with a string.
3. Place the string across the pulley so that the weight can hang freely from
the end of the board.
4. Adjust the hanging weight (mw) so that the block (when started) will move
at a CONSTANT SPEED (Very important).
5. Do several trials for loads of 0, 100, 200, 300, and 400g weights on the
block, and determine friction coefficient for each load.
6. Repeat the procedure with a block that has leather on the bottom.
7. Repeat the procedure with a block that has sand paper on the bottom.
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Answer the following questions and create the graphs and include
them on your lab report.
In addition to data collected and calculated (in the chart) answer next questions and create
graphs.
1. Why is the pulley necessary?
2. Why is coefficient of friction constant or why not?
3. Derive the formula µ = Ff/N = mw/m
4. Graph µ = f(m) for each case. Show values for µ on the vertical axis and values for m
on the horizontal axis. You should use Excel to create these graphs.
5. Graph µ = f(mw) for each case. Show values for µ on the vertical axis and values for
mw on the horizontal axis. You should use Excel to create these graphs.
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EXPERIMENT NO.5: FRICTION ON AN INCLINED PLANE
Teaching Element:
When a block is placed on an incline the tendency is for the block to slide down the
plane. If the angle of inclination is small the block is prevented from slipping by the
friction between the surfaces. As the angle is increased, the force exerted down the plane
due to the weight of the block also increases, but the force pressing the surfaces together
decreases. At the Angle of Friction, the force acting down the plane just overcomes the
friction and sliding takes place.
Student Objectives:
To investigate friction on the inclined plane and to investigate the relationship between
the required forces (applied parallel to the plane) to slide a block up the plane.
Apparatus
Friction plate, slider, pulley, cord, weights, spring balance.
Friction on Inclined Plane
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Method
1. Place the slider and weights on the friction plane, and attach the spring balance as
shown in the diagram.
2. Gradually pull the spring balance and note the force registered just as motion
commences.
3. Keep the slider moving with uniform speed and again observe the reading on the
spring balance.
Conclusion
Is it more difficult to start a body moving or to keep it moving?
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EXPERIMENT NO.6: ANGLE OF FRICTION
Teaching Element:
When a block rests on a horizontal plane its WHOLE weight presses on the plane and the
pressure between the surfaces sets up a resistance to movement which is called friction. If
the plane is vertical no pressure takes place between the surfaces because the whole
weight is acting downwards parallel to the plane and the block will slide down the plane.
Therefore, when the plane is inclined at an angle between the horizontal and vertical,
PART of the block weight acts parallel to the plane and PART of the weight produces
pressure between the surfaces.
As the angle of inclination increases, the force acting along the plane increases but the
force pressing the surfaces together decreases and so the friction force decreases. At a
certain angle the force acting down the plane will overcome the frictional resistance
between the surfaces and sliding will take place. The angle at which sliding begins to take
place is called the ANGLE OF FRICTION and this experiment will show the relationship
which exists between this angle and the Coefficient of Friction.
Student Objectives:
1. To measure the Angle of Friction and from it find the Coefficient of Friction.
2. To show that the Coefficient of Friction is equal to Tangent of the Angle of Friction.
Object
To determine the angle of friction for various materials in contact, and to find the
connection between angle of friction and coefficient of friction.
Apparatus
A plane as shown in sketch, sliders as used in Experiment No.4
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Friction on Inclined Plane
Theory
The ratio F/N mentioned in Experiment No.4 is usually denoted by µ.
In the above diagram tan θ
= H/L
Method
1. Place a slider on the plane.
2. Tilt the plane, until the slider just moves.
3. Measure the height “H,” the corresponding length “L,” and so find tan θ and θ (the
angle of friction).
4. Repeat for each slider.
Observations:
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Conclusions:
1. Compare the values of tan θ with the values of µ for the same materials.
Is tan θ = µ?
2. Does the area of contact have any effect on the force of friction?
3. Does the ratio F/N vary with different materials or is it a constant for all materials?
4. Now state the laws which you have obtained from each experiment on friction.
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EXPERIMENT NO.7: SLIDING FRICTION
Teaching Element:
When two rough surfaces are made to slide over one another, the minute, uneven surface
particles resist the sliding and are sometimes torn away. This resistance to sliding is
called friction. Even so-called "smooth" surfaces have microscopic roughness which
causes friction and the friction force must be overcome before sliding can take place.
Friction is usually regarded as wasteful, as in machines where it absorbs power and
causes wear, but it can be useful, for example in friction brakes. In designing machines
where sliding takes place, the effect of friction must be taken into account and for this the
LAWS OF FRICTION are used. The laws are only approximately true, but they form a
useful and practical basis for dealing with friction problems.
Friction opposes sliding and depends on the roughness of surfaces in contact. In practice
it is found that the friction force is a fixed proportion of the force pressing the surfaces
together. This proportion is called the COEFFICIENT OF FRICTION. Friction which
opposes movement from rest is called STATIC FRICTION. As soon as sliding takes
place it is found that less force is required and this is called KINETIC FRICTION.
Student Objectives:
To verify the Laws of Friction and to measure the Coefficient of Friction for different
materials.
Apparatus:
Friction plane, slider, pulley, cord, weights, scale-pan.
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Friction on Inclined Plane
Method:
1. Set up the apparatus as shown in above
2. Place a weight on the slider. Note the reaction, N lb (weight of slider and added
weight.)
3. Add weights to the scale-pan, until motion commences. Note the force, F lb., needed
to cause motion.
4. Calculate the ratio F/N.
5. Repeat the experiment for several loads.
6. Plot the F-N graph
Observations:
Conclusions:
1. Is the graph a straight line passing through the origin?
2. What conclusion may be drawn regarding the ratio F/N?
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3. What is the ratio F/N called?
4. Does the ratio F/N vary with different materials or is it a constant for all materials?
5. Now state the laws which you have obtained from each experiment on friction.
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EXPERIMENT NO 8: PRINCIPLE OF MOMENTS
Teaching Element:
When forces produce a turning effect, this turning effect can be measured by the product
of the force and the perpendicular distance between the pivot and the line of the force.
The product is called the TURNING MOMENT of the force.
If a body has several forces applied to it which have turning effects in opposite directions,
the body will not turn if the total turning moments in each direction are equal. This is
called the PRINCIPLE OF MOMENTS.
The Principle of Moments is frequently used in engineering and building work where
forces have to be balanced to prevent any turning movement. It can be applied both to
parallel forces and to oblique forces; but in all cases, when calculating the turning
moment the length is the perpendicular distance from the pivot to the line of the force. A
method of calculating the effect of turning forces to produce equilibrium is to say The
Moments Clockwise = The Moments Anti-Clockwise.
Student Objectives:
The object of this experiment is to verify the Principle of Moments for parallel and nonparallel forces.
Apparatus:
Two spring balances, several weights, cord.
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Beam Forces
Theory:
Vertical component = AB sin ά
Horizontal component = SB cos ά
Method:
1.
2.
3.
4.
Arrange the apparatus as in Fig.
Pull cord A, making sure that the angle between A and C is 900.
Note the angle ά, the weight W, and the spring balance readings SA and SB.
Find the horizontal and vertical components of SB graphically and by calculation.
Conclusion:
Do the graphical and calculated and calculated values agree with the values found by
experiment?
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EXPERIMENT NO 9: THE PIVOT (OR BEAM) BALANCE
Teaching Element:
Experiment No 5 shows that if a pivoted bar has forces applied to it which have turning
effects, the body will not turn if the turning moments in each direction are equal. A
turning moment being the force multiplied by the perpendicular distance from the centre
of the pivot.
The pivot (or beam) balance makes use of this principle for weighing. In the beam
balance, the weight to be measured is placed in one pan and is balanced by known
weights in the other, both pans being at the same distance from the pivot. In the slide
balance, the arms are of unequal length; the weight being measured is placed in the pan
on the short arm and balanced by a known weight which slides along the long arm. A
scale marked on the long arm is calibrated to show the weight in the pan.
Student Objectives:
The object of this experiment is to demonstrate that the action of weighing with a beam
balance or slide balance is based upon the Principle of Moments.
Apparatus:
Wooden beam, two spring balances, various small weights.
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Beam Forces
Method:
1. Suspend the beam from the two spring balances as shown in Fig.
2. Before placing any loads on the beam, note the reading on the spring
balances. (Let these be P1 and Q1 lb. Respectively.)
3. Place some weights on the beam.
4. Read the spring balances. (These values are P and Q lb.)
5. The differences (P-P1) and (Q-Q1) give the reactions on the supports due to
the added weights.
6. Calculate the reactions on the supports.
7. Repeat the experiment for several different loadings.
Conclusions:
1. Do the calculated values agree with the observed values?
2. What do you notice about the sum of the upward forces and the sum of the
downward forces?
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EXPERIMENT NO.10 FORCES IN FRAME STRUCTURES
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Experiment No. 11 Finding the point where an Object’s mass acts
Topic
Center of gravity and equilibrium
Introduction
The weight of a body is the force that the body exerts down towards the Earth,
to which it is attracted by gravity. The center of gravity of a body is the fixed
point through which all the weight of the body appears to act. In this
experiment, you will determine the center of gravity of some flat shapes and
show that they will balance if supported at this point. You will also investigate
the stability of objects and show how an object will remain stable (in equilibrium) if its center of
gravity is supported over its base.
Time required
Part A: about 10 minutes per shape
30 minutes for Part B
Materials
For Part A:
number of shapes (e.g., triangle or rectangle) cut from light colored cardboard,
each having 4 – 5 holes punched around the edges (shapes can be regular or irregular, with
straight or curved sides, and a longest dimension of about 15 – 20 cm) support stand and clamp 1
meter fine string or thick thread small weight (e.g., a large nail) knitting needle pencil 1 meter
rule
For Part B:
4 bottle corks (identical)
4 toothpicks
6 toothpicks cut in half to make 12 sticks of equal length and pointed at one end
small board (about 20 ⋅ 30 cm) such as a cutting board
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Procedure
Part A: Determining the center of gravity of a flat shape
1. Secure the needle in the clamp as shown in diagram 1 below. The clamp
should be about 45 cm above the surface of the bench or table.
2. Tie the small weight to one end of the fine string or thick thread to form a
plumb line. (When suspended, a plumb line points directly towards the Earth’s
center of gravity and thus shows the vertical line.) Make a loop at the other
end of the string so that the plumb line is about 30 cm long.
3. Select one of the cardboard shapes. Insert the needle through one of the holes
in the shape and secure the plumb line around the needle as shown in
diagram 2 above.
4. Use the pencil to make a series of dots marking the line taken by the plumb
line on the surface of the cardboard shape.
5. Remove the plumb line and shape from the needle. Connect the dots showing
the position of the plumb line to make a line (see diagram 3 below).
6. Repeat steps 3 to 5 using all the holes on the cardboard shape. You will then
have a series of lines as shown in diagram 3 below.
7. Attempt to balance the shape on a finger placed at the point where the lines
intersect (see diagram 4 on the next page).
8. Repeat steps 3 to 7 for each cardboard shape.
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Part B: Equilibrium
1. To make object A, carefully push four of the short sticks into one of the corks
as shown in diagram 5A below. The four sticks should enter the cork to a
depth of about 5 mm and point directly downwards.
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2. To make object B, carefully push the four long sticks into another cork as
shown in diagram 5B above. Position them in the cork as you did for object A.
3. To make object C, carefully push four of the short sticks into a third cork as
shown in diagram 5C above. The four sticks should enter the cork to a depth
of about 5 mm and splay out.
4. To make object D, carefully push four of the short sticks into the fourth cork
as shown in diagram 5D above. The four sticks should enter the cork to a
depth of about 5 mm and angle in.
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Analysis
Part A: Determining the center of gravity of a flat shape
1. Do the lines drawn on each cardboard shape meet at a single point?
2. What happens if the shape is supported at this point?
Part B: Equilibrium
1. Where would you estimate the center of gravity of the objects to be?
2. How do you relate the position of the center of gravity to the order in which
the objects toppled?
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EXPERIMENT NO.12 . M O R E S C I E N C E O F B A L A N C E
For this week’s experiment, I thought we would take the Science of Balance a
bit further. We saw in the first experiment that for an object to balance, you
had to have the center of gravit y direct ly above the base. This t ime, we are
going to reverse that to see that you can also balance an object by having the
center of gravit y direct ly below its base.
To try this, you will need:
1.
2.
3.
4.
5.
2 forks
a piece of apple, potato or other firm vegetable about 2 inches square
a wooden toothpick or match st ick
masking tape
a marker or ink pen
WStart by st icking a fork into one side of the piece of apple. Then st ick the
other fork into the other side. You want them both angled downward slight ly,
to form a large “V” shape, wit h the apple at the point of the V, and the forks
forming the two arms.
Stick the toothpick into the apple, in between the two forks, point ing in t he
same direct ion as the handles o f the forks. Now you have a “W”, with the
toothpick forming the center point.
Place the po int of the toothpick on your finger, and try to balance the forks.
It works! The whole thing will balance quit e easily. How?
In our previous experiment, we balanced by keeping the center of gravit y
direct ly o ver the base. Here, there is nothing over the base, but then there is
nothing below the base eit her. This is a case where an object’s center of
gravit y is outside the object. While it may sound strange, it is more commo n
than you might think.
We can find that center of gravit y wit h some masking tape and a pen. Balance
the forks on your finger again. You want to st ick a strip of masking tape fro m
one fork to the other, so that it passes directly under the po int where the
toothpick is balanced on your finger. Then make a mark on the tape, direct ly
under that balance po int. That mark shows you the center of gravit y for the
object.
Now push the toothpick into the apple unt il only about half an inch st icks
out. Again, balance it on your finger. The angle will be very different, but
you will find that the mark you made on the tape is st ill direct ly under the
balance point.
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Remove the toothpick and st ick it into the apple in a different spot. Try
balancing it again. If you can get it to balance, you will find that the mark is
again underneath the balance po int.
Take some t ime to play wit h this experiment. It is fun to see what you can
balance it on, including a string stretched between two chairs, if you have a
steady hand. Use it to impress your friends. Then tell them the science behind
it and impress them again.
Have a wonder-filled week.
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Experiment 13 to verify the laws of limiting friction
Take a block of wood of specific mass, a thread, pulley, a pan and a few weights and arrange them as
shown in the figure.
Now add a few weights in the empty pan. The block does not move. This shows that even though the string
pulls the block to the right, the frictional force pulls it to the left. Hence, 1st law is verified.
Since the frictional force f acts horizontal to the surface, it is tangential to the surface of contact. Hence, 2nd
law is verified.
Keep adding weights in the pan and on the block so that the block just begins to move. Now add some
additional weight on the block and adjust the weight on the pan so that the block just begins to move again.
Note the weight of the block + the weight on it and the weight on the pan. You will notice that they increase
or decrease proportionally. Hence, law three is verified.
Replace the wooden block with a glass block, stone block and note the weight of the pan. This observation
will verify the law of static friction i.e. the fourth law. Now consider any one of the blocks. Change the face of
the surface of contact and position of the block. You will notice that the weight in the pan will be the same for
all cases (when the block just begins to move). This verifies the law of limiting friction. This verifies the fifth
law.
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EXPERIMENT NO.14 TO VERIFY LAMI’S THEOREM
The simple arrangement used to verify the Lami’s theorem is generally called
parallelogram law apparatus as shown in Fig. 1
Procedure
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EXPERIMENT NO.15 TO VERIFY TRIANGLE LAW
The simple arrangement used to verify the triangle law is generally called
parallelogram law apparatus as shown in Fig. 1
PROCEDURE
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