Statics Lab

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UNESCO-NIGERIA TECHNICAL &

VOCATIONAL EDUCATION

REVITALISATION PROJECT-PHASE II

YEAR I- SE MESTER I

PRACTICALS

Version 1: December 2008

NATIONAL DIPLOMA IN

MECHANICAL ENGINEERING TECHNOLOGY

MECHANICAL ENGINEERING

SCIENCE[STATICS]

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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

Week 3

3. Experiment No 3: Polygon of Forces

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 ie nc e o f ba la nc 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 arrow-

head 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 arrow-

head 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. Is BC parallel to the cord carrying load R?

2. Does BC represent the force R drawn to scale?

3. Can the forces be represented by the sides of a triangle?

4. What do you notice about the arrow-heads on the forces in the triangle ABC?

5. 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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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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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 non-

parallel 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. Arrange the apparatus as in Fig.

2. Pull cord A, making sure that the angle between A and C is 900.

3. Note the angle ά, the weight W, and the spring balance readings SA and SB.

4. 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 . MORE SCI E NCE O F B AL AN CE

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 gravity 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 gravity direct ly below its base.

To try this, you will need:

1. 2 forks

2. a piece of apple, potato or other firm vegetable about 2 inches square

3. a wooden toothpick or match st ick

4. masking tape

5. 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, with 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 the

same direct ion as the handles of the forks. Now you have a “W”, with the

toothpick forming the center point.

Place the point of the toothpick on your finger, and try to balance the forks.

It works! The whole thing will balance quite easily. How?

In our previous experiment, we balanced by keeping the center of gravity

direct ly over the base. Here, there is nothing over the base, but then there is

nothing below the base either. This is a case where an object’s center of

gravity is outside the object. While it may sound strange, it is more common

than you might think.

We can find that center of gravity with some masking tape and a pen. Balance

the forks on your finger again. You want to st ick a strip of masking tape from

one fork to the other, so that it passes directly under the point where the

toothpick is balanced on your finger. Then make a mark on the tape, direct ly

under that balance point. That mark shows you the center of gravity 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 point.

Take some t ime to play with 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

Statics Lab

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