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Physics Department

Electricity and Magnetism Laboratory

CIRCULAR MOTION

1. Aim

The aim of this experiment is to study uniform circular motion and uniformly accelerated

circular motion. The motion equation for a rotating rigid body will be studied, and a moment of

inertia will be calculated.

2. Overview

Circular motion is a curved motion where the path (or trajectory) is a circumference, e.g.

the motion of any point on a rotating disk or wheel. On a first approximation, the motion of the

Moon around Earth and that of an electron around a proton in a hydrogen atom are circular

motions. Due to the Earth's rotation, all bodies on its surface are on circular motion around the

axis of rotation of the Earth.

2.1 Uniform circular motion equations.

In circular motion with constant speed, the velocity vector, ,v is tangential to the

circumference (as velocity is always tangential to the path). Distance covered, s, is always

measured along the path, which in this case is an arc of circumference.

Equations are similar to those of uniform rectilinear motion (constant velocity), but instead of

distance covered s there is the angle swept out , and instead of linear velocity ,v angular

velocity, , (angle swept out per time unit). See Figure 1.

Rv [1]

0

dt

d [2]

t 0 [3]

R is the distance from the body to the axis of rotation.

In this motion, the velocity vector v has a constant modulus but variable direction and

sense. This implies that there is acceleration: normal acceleration

R

vaN

2

directed towards the axis of rotation.

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Figure 1.

2.2 Uniformly accelerated circular motion equations.

When the angular velocity of a particle in circular motion changes over time, angular acceleration is defined as

dt

d [4]

The equations obtained are:

t 0 [5]

2

002

1tt [6]

Aside from [1] and the normal acceleration, now there is also a tangential acceleration,

RaT (See Figure 2).

Figure 2.

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2.3 Relationship between moment of force and angular acceleration.

Newton's second law of motion indicates the relationship between the forces applied onto a

body and the resulting acceleration amF , [7]. Now we are studying circular motion, and

instead of force we speak about moment of force, also called torque. The moment of a force

with respect to a certain axis is the product of the force and its distance to the axis,

FRM (see Figure 3). This torque does not cause displacement but rotation, by causing

angular acceleration.

In a similar fashion as [7] but changing force for moment and acceleration for

angular acceleration, we arrive to IM , [8], where I is the moment of inertia, which

depends on the mass distribution and the geometry of the body, and has dimension 2ML (in

the case of a coin it is 2

2

1mRI ).

Figure 3

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3. Equipment

Figure 4.

1. Photogate.

2. Rod support.

3. Rod.

4. Power source.

5. Motor.

6. Wires.

7. Pulley, fixed to the edge of the table.

8. Weight support.

9. Weights.

10. String.

11. Ruler.

12. Vernier caliper.

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4. Experimental Procedure

4.1 Uniform circular motion.

The experimental set-up is shown in Figure 5.

Connect the motor to the power source. Turn on the power source and set a voltage so that

the speed of rotation of the rod is as small as possible but stable. Never go beyond 1.5A for

the current or the motor might be damaged!

Now we are going to measure the linear speed of the rod (at the position of the photogate)

and the angular velocity. The procedure will be:

Learn how to use the photogate (sec. 4.3).

To obtain the linear speed, use the photogate measure t2 (time that takes for the end

of the rod to go through the photogate) three times.

To obtain the angular velocity, use the photogate measure t3 (time between two passes

of the rod's ends, which corresponds to a 180º angle swept out) three times.

This procedure has to be done for 5 different velocities of the rod. In order to set new

velocities values, increase the voltage of the power supply. The velocity values chosen has to

be:

Different enough so the values t2 and t3

change their values on the display.

Low enough so the current is never higher than 1.5A.

Figure 5.

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4.2 Uniformly accelerated circular motion.

Prepare the set-up shown in Figure 6. Note that the motor is not required for the set-up.

Wind up the string around the rod support so that when dropping the weight you will see the

rod start to spin increasingly fast. Ensure that the rod makes several turns before stopping

when the string is unwound. r Is the radius of the cylinder where the string is wound up.

Set a 10 g weight on the weight support.

Now we are going to obtain experimentally the relation between the linear velocity v

of the rod position where the photogate is placed and the angle swept . Later we will

compare the result with the expected using the theory (see appendix 5.1). We are going to

measure the linear velocity for 5 different angular positions 2/ , 2/3 , 2/5 ,

2/7 y 2/9 . In order to obtain the linear velocity for every one of the angular

positions we will measure with the help of the photogate the time 2t . The linear velocity v is

obtained dividing the width of the rod by 2t . We will follow the next procedure:

Select the photogate measure mode 2t .

To measure on 2/ set the rod at a 90º angle from the photogate, as shown in

Figure 6. Activate the photogate trigger (set button) and release the rod. Thus, the rod

will start moving in uniformly accelerated circular motion. When the end of the rod goes

through the photogate, the angle swept out is 90º. In that instant the sensor

measures 2t , and from this, linear velocity )2/( v can be calculated.

To measure on 2/3 angle is obtained when the opposite end of the rod goes

through the photogate. The set-up of the previous measure is repeated (wind up the

stricg around the rod and hold the rod with your hand) but we don’t activate the

photogate trigger. Release the rod and it will start spinning and, after the first rod

end passes through the photogate, you have to activate the trigger. Thus, when

the second rod end passes through the gate, a measure of 2t will be taken when the

displacement is 2/3 .

Repeat the process for the other angles, taking into account that the gate has to be

triggered just before the angle we want to measure. For example, if we are measuring

on 2/9 , the rod has to pass through the gate four times. After the fourth pass

you should activate the gate and it will measure the fifth pass (that correspond to the

desired angle). Of course, four passes of the rod correspond with two passes of one end

of the rod and two passes of the other end of the rod!

Obtain one single measurement of each angular position and fill in the table with the values,

the measured 2t , and the computed v.

Repeat the procedure using the 20g weight.

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Figure 6.

4.3. Photogate.

Figure 7.

Check that the sensor is correctly plugged.

To measure 2t set the switch (bottom-right on the figure) on the second position (next to

Count). Measure the time interval in seconds that takes the rod to go through the sensor.

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Reset the count with the Set button.

To measure 3t set the switch in the third position.

The sensor starts counting when one end goes through, and stops when the other end goes

through. So, 3t is the time taken in sweeping a 180º angle.

5. Appendix

Relationship between linear velocity and angle swept out in uniformly accelerated

circular motion.

Motion starts at rest. In wRv we can substitute tw which yields:

Rtv . From [6] /2t and substituting in we arrive to:

Rv 2 [9]

Take the logarithm from both sides:

Rv lnln2

1)2ln(

2

1ln [10]

5.2 Relationship between moment of force and angular acceleration.

Figure 8.

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(See Figure 8). From equation [8] and bearing in mind that the only force present is the

tension of the string,

IrT [11]

To obtain the tension of the string, we apply [7] on the weight holder.

amTgm portaporta [12]

From ra we substitute in [11] and the moment of inertia results:

2rm

grmI porta

porta

[13]

The moment of inertia is always positive, 0I :

gr [14]

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Relationship between linear velocity and angle swept out in uniformly accelerated

circular motion.

Motion starts at rest. In wRv we can substitute tw which yields:

Rtv . From [6] /2t and substituting in we arrive to:

Rv 2 [9]

Take the logarithm from both sides:

Rv lnln2

1)2ln(

2

1ln [10]

5.2 Relationship between moment of force and angular acceleration.

Figure 8.

(See Figure 8). From equation [8] and bearing in mind that the only force present is the

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tension of the string,

IrT [11]

To obtain the tension of the string, we apply [7] on the weight holder.

amTgmportaporta

[12]

From ra we substitute in [11] and the moment of inertia results:

2rm

grmI

porta

porta

[13]

The moment of inertia is always positive, 0I :

gr [14]

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