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#### UNIFORM CIRCULAR MOTION LAB Objective: In this experiment we will determine the centripetal force acting on an object undergoing UCM and determine its relationship to other parameters such as mass, radius, and velocity

###### Physics

UNIFORM CIRCULAR MOTION LAB

Objective:

In this experiment we will determine the centripetal force acting on an object undergoing UCM and determine its relationship to other parameters such as mass, radius, and velocity.

Available Equipment:

Circular motion apparatus, stopwatch, mass hanger, lab masses, broom, and bowling ball

Theory: 4

Newton’s first law states that an object will remain in motion with a constant velocity unless acted upon by a net external force. When a net force acts on an object, the velocity of that object changes, therefore the object accelerates. Since velocity is a vector, it has both magnitude (the speed) and direction. If either the magnitude or the direction of the velocity changes, the object accelerates. The direction of the velocity vector is always tangent to the trajectory and thus when an object travels in a circular path, the direction of the velocity constantly changes. If the velocity changes, the object accelerates, and therefore there must be a net force acting on that object. Newton’s second law states that Fnet = ma and from this equation one can see that the net force has the same direction as the acceleration.

An object undergoes uniform circular motion (UCM) when it moves with constant speed in a circle.

The acceleration of an object in UCM is the result of the change only in the direction of the velocity vector, it is always oriented toward the center of the circle and it is called centripetal (or radial) acceleration.

Therefore, the net force acting on an object in UCM is oriented toward the center of the motion and it is also called centripetal force. This is not a new force, it is the net force (or the sum of all forces) acting on the object.

An object of mass M following a circular path of radius R with a constant speed v experiences a centripetal (or radial) acceleration given by acp = v2/R and a net force of Fnet = macp = m. v2/R oriented toward the center of the circle. If the object completes a given number of revolutions N in a total time t, then the period of motion is given by T = total time/number of revolutions = t/N. Since the path is circular and the speed is constant, one can calculate the speed of the object as the total path traveled (the circumference of the circle) in one revolution: v = 2πR/T.

In this experiment, an object of mass M is suspended by a string and connected to a horizontal spring, as shown in the figure below. When the entire apparatus rotates with constant speed, the mass M executes a UCM. The elastic force in the spring applies the centripetal force to the mass M and forces it to follow a circular trajectory. If the mass M is spun just right, the string is perfectly vertical and thus the tension force in the spring is balanced by the weight of the mass M, and therefore the only force left unbalanced is the elastic force, which plays the role of net force, or centripetal force in this case. A schematic representation of the rotating apparatus and a free body diagram are provided below.

M = supporting mass

A = arm

W = weight

E = mounting eyelet

S1, S2 = points at which the spring is attached

d = stretching distance

P = post

Schematic representation of the apparatus

Free body diagram for the rotating mass M

F tension

F elastic

F Weight

Ftension = Fweight => Ftension = mg

Felasyic = Macp = Felastic = M* v2/R

EXPERIMENTAL PROCEDURE:

As you follow the experimental procedure and collect data, enter the data in the Data and Analysis Section.

BE SURE TO WEAR EYE PROTECTION.

1. Examine the string supporting mass M. If it is worn or frayed, replace it.

2. Decide what radius you want for your circular path. The radius can be measured using the scale on the bottom of the apparatus. For the first trial, an Intermediate value is recommended, for example

16-18 cm.

3. With the spring unhooked, adjust the length of arm A so that VM hangs directly over the chosen value of R. The string that holds the mass M exerts only a vertical force on M, which counterbalances the gravitational force, Mg.

4. The height of mass M should be checked so that the two points where the spring attaches (S1 and S2) are at the same vertical height. If not, the position of the mounting eyelet, E, can be changed by its locking screw adjustment. This ensures that the spring will be horizontal and the force it exerts will also be horizontal.

5. When the spring is hooked at S1 and S2, it will need to stretch in order for M to hang over the chosen value of R. The position of S1, or equivalently the distance d, will determine how much stretch is needed. Initially you might adjust the S1 screw to a mid range setting.

Activity 1: Determine the elastic force acting on the mass M from uniform circular motion (rotating)

6. The string that suspends mass M is not in a vertical] position now.

7. Rotate the post, slowly at first, then faster, until mass M hangs over to the desired radius R. Be careful not to over spin and allow mass M to strike the pulley.

8. Maintain a steady speed so that mass M hangs over the desired radius R by periodically applying a little “tweak” with your fingers. (The large counter weight W is present to increase the rotational inertia of the apparatus so that once the proper speed is attained it is easy to maintain it.) Keep your eyes in the plane of motion, but far enough away so that the rotating mass does not hit you. BE SURE TO WEAR EYE PROTECTION.

9. Have your lab partner time the motion for a large number of revolutions (50 revolutions for example).

Then switch job assignments and repeat the measurement (your instructor may require you to use the average value obtained from taking several measurements for improved accuracy).

10. Do not adjust any of the parameters on the apparatus. Move on to Step 11.

Activity 2: Determine the elastic force acting on mass M by balancing forces (stationary)

11. Based on your result of the elastic force from Activity 1, calculate the expected mass m that will be necessary to hang off the pulley to make mass M hang vertically. Record your calculation in the Data and Analysis Section.

12. Attach one end of a light string to mass M, pass it over the pulley, and attach a mass hanger to the other end of the string (see the figure below). Add lab masses to the mass hanger (combined mass m) until mass M hangs over the desired value of R.

The force diagram is shown in this figure.

For object M:                 in the vertical direction Ftension = Fweight = Mg

In the horizontal direction Ftension2 = Felastic       Felastic = mg

For hanging mass m:    in the vertical direction Ftension2 = Mg

13. Repeat the experiment, Steps 6-12, for configurations 2-3. (Only adjust the radius R, do not adjust the length of the screw d)

14. For configuration 4, while keeping the radius, R, set at 18cm, adjust the length of the screw, d.

Uniform Circular Motion Lab: Data and Analysis Section

After collecting your data, fill out the questions below. Do not forget to include units for each one of the values written.

Activities 1&2: Determine the elastic force acting on mass M from uniform circular motion and by balancing forces

Start with Configuration 1, Perform Activity 1 and Z, fill in the tables below, then move on to configuration 2 and continue the measurements.

Mass M

Show all of your calculations in the data table

 Configuration 1 Radius R Activity 1 # of revolutions N Time for N revolutions Period T Speed v Centripetal Accel., acp Centripetal force Fcp Elastic force Felastic 16 cm 50 46.7s Activity 2 Calculated hanging mass                    m Measured Hanging mass, m Weight of measured hanging mass Elastic force Felastic 290g 2.8N 2.8N (Don’t change the length of screw d) Configuration 2 17 cm Activity 1 # of revolutions N Time for N revolutions Period T Speed v Centripetal Accel., acp Centripetal force Fcp Elastic force Felastic 50 36s Activity2 Calculated hanging mass                    m Measured Hanging mass, m Weight of measured hanging mass Elastic force Felastic 490g 4.8N 4.8N (Don't change the length of screw d) Configuration 3 18 cm Activity 1 # of revolutions N Time for N revolutions Period T Speed v Centripetal Accel., acp Centripetal force Fcp Elastic force Felastic 50 31s Activity 2 Calculated hanging mass                    m Measured Hanging mass, m Weight of measured hanging mass Elastic force Felastic 690g 6.8N 6.8N

*For configuration 4, keep the radius R at 18cm, change the length of the screw, d.

 How did you the length               The length of d increased      The length of d decreased Of the screw d? (circle one) Configuration 4 *18 cm Activity 1 # of revolutions N Time for N revolution s Period T Speed v Centripetal Accel., acp Centripetal force Fcp Elastic force Felastic 50 27.5s Activity 2 Calculated hanging mass                    m Measured Hanging mass, m Weight of measured hanging mass Elastic force Felastic 920g 9.0N 9.0N

1. For each configuration, examine the elastic forces determined from each activity. How do the elastic forces from Activities 1 and 2 compare?

a. Explain why they would be similar.

b. Explain why the elastic forces would differ between activities.

2. (a) Look at the apparatus. Predict How the elastic force (centripetal force) would change if the radius R is changed while the length of the screw d is fixed?

(b) Look at your results in Configurations 1, 2, and 3, do they agree with your prediction in (a)?

If they don’t agree, you need to reconcile your prediction and your results.

3. (a) Look at the apparatus. Predict How the elastic force (centripetal force) would change if the length of the screw d is changed while the radius R is fixed?

(b) Look at your results in Configurations 3 and 4, do they agree with your prediction in (a)?

If they don’t agree, you need to reconcile your prediction and your results.

4. In Activity 2, how did the measured hanging mass m compare with the calculated values?

Calculate the percentage difference = ? calculated value - measured value|/1/2(calculated value + measured value) x 100% for each configuration.

5. In determining the elastic force in Activity 2, only one experimental quantity is involved, the hanging mass m. Its value in your experiment has uncertainty. Uncertainty is also called error but not mistake. Uncertainty is due to the limit to the accuracy of experimental results.

a. On your last configuration of Activity 2, from your obtained hanging mass m value, add 1 gram to the hanger, do you see visible impact on the motion of the object M?

b. If not, keep adding a few grams till you observe visible impact on the motion of the object M.

The added a few grams is the uncertainty of the hanging mass m. Write down the result.

Uncertainty of hanging mass m =

c. How was the value of mass M measured? What is its major source of uncertainty? What is its estimated uncertainty?

d. Should the uncertainties in questions b and c above be the same?

6. Examine the experimental quantities that were involved in determining the elastic force in

Activities 1 and 2. Which method contains greater uncertainty? Explain why.