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Lab Exercise 5: Centripetal Acceleration

• Follow the instructions and directions below for this lab.  Disregard the outline in the manual for your LabPaq Kit.

• Read this document entirely before starting your work.

• Do not forget to record your measurements and partial results.

• Submit a Laboratory Report through Moodle, as shown in the last section of this outline.  Remember that the Laboratory Report should include the answers to the questions below.

GOALS

(1) To calculate the angular velocity of a spinning object using varying hanging and rotational masses and varying radii  

(2) To calculate the theoretical centripetal force;

(3) To calculate the experimental centripetal force.

INTRODUCTION

When objects, such as a carousel, move in a uniform circular motion, they are moving at a constant speed, while their direction of velocity is changing. The word centripetal means center seeking. When acceleration of a circular moving object is directed toward the center, the acceleration is centripetal and the acceleration is called centripetal acceleration.

Newton’s first and second laws of motion state that an object moves at a constant speed in a straight line unless an external force acts upon that object and that a force causes an object’s acceleration. By following theses laws, the force on a circular moving object is called centripetal force. Centripetal force accelerates an object by changing the direction of its velocity without changing its speed.

Mathematically, centripetal acceleration is represented as:

with ac being the centripetal acceleration, v the velocity and r the radius of the circle.

The centripetal force, in turn, can be represented as:

with Fc being the centripetal force and m the mass of the object.

An example of centripetal acceleration is the Earth/Moon relationship. Earth and the Moon exert gravitational forces on each other and the Moon undergoes centripetal acceleration toward the center of Earth.

PROCEDURE

In this experiment, a rubber stopper is connected to a string and is rotated in a horizontal circle. The tension in the string causes the stopper to undergo centripetal acceleration.

Rotational Velocity

The period of revolution or period—the time it takes for the object to complete one revolution— is represented by T (this is similar to the notation we used in the previous laboratory with the pendulum apparatus). The speed v of the rotating object is calculated by dividing the circumference of the circle of radius r (2πr) by T. This velocity can be referred to as rotational velocity or angular velocity.

Therefore, to determine the constant velocity of a rotating object, we need to measure the time T required to make one revolution using the following equations:

In addition to centripetal acceleration, the force of gravity acts on the rubber stopper as it is whirled along a horizontal plane. Because gravity acts perpendicular to the centripetal force, the orbital plane of the rotating mass lies below the horizontal plane at the top end of the vertical tube. Despite these factors, the data obtained from this experiment should be reasonable approximations that demonstrate the basic relationships among the variables.

Constant mass, variable radius.

In this section we will investigate the effects of changing the radius of the system on the centripetal force.

Choose an area that is free from obstructions and breakable objects. You will be swinging weights on a string and if these weights break free, they could potentially hit objects or people. Choose an area where only your assistant is present to reduce the risk of people being injured.

Wear goggles so that the rotating stopper does not hit your eyes.

Record the number of washers from your kit in Table 1. Place all of the washers into a bag to weigh their mass and record the total mass in Table 1. Find the average mass of each washer in kilograms and record it in Table 1.  Also weigh the mass of the rubber stopper.

Pull out the 4.0 m of string provided in your apparatus kit.

Tie a the 4m string to a rubber stopper (the rotating mass), slide the string through a glass cylinder, and tie the string to our hanging mass. Before threading the string through the glass rod, make sure the smooth end of the glass rod is at the top nearest the rotating rubber stopper.

Thread about 30 g of washers onto the end of the string opposite from the stopper. Record this constant hanging mass.  User paper clips to ensure that the washers do not fly away.  If needed, open up the paper clip to secure the washers. Figure 3 shows a detail of this.

QUESTION 1

What is the actual (measured) mass of the washers?

Figures 1, 2 and 3 show the experimental setup.

Figure 1: Experimental setup

Figure 2: Another picture of the experimental setup.

Figure 3: Details of Washers at one end.

Tie another paper clip about 20 cm above the washers. When finished, your apparatus should look like the one in Figure 4.

Figure 4: Experimental setup

Pull the string through the glass rod so that approximately 0.7 m of string is between the glass rod and stopper. Practice swinging the stopper around in a circle over your head as shown in Figure 5 while holding onto the glass rod. Support the suspended mass containing the washers with one hand and hold the rod in the other. Be careful and review the safety precautions at the beginning of this procedure.

Figure 5:  Student working in the experiment (Picture courtesy of Chad Saunders, TESU student)

Swing the stopper in a circular motion. Slowly release the hanging mass and adjust the rotating speed of the stopper so that the paper clip attached to the string above the washers stays a few centimeters below the bottom of the tube, neither rising nor falling.

Do not move your hand too much while swinging the stopper. Ideally, the steel washers should be stationary. Keeping your hand steady will help the rubber stopper move smoothly. Practice stopping the spin while simultaneously grasping the string just above the tube. This action will allow you to measure the radius of the spin circle, which is the length of the string from the top of the tube to the center of the stopper.

Stop spinning the rubber stopper and use the measuring tape to measure the length of the string in meters. This is the length of the string between the glass tube and rubber stopper.

Record this length as the radius for Radius 1 in Table 2.

Once you are able to spin the stopper with a steady pace, you can begin the experimental portion of the lab.

As you continue with the experiment, complete the appropriate cells in Table 2.  Note the following:

• To estimate the time for 1 revolution, divide the time for 10 revolutions by 10.

• Use the radius in each row to calculate the length of the circumference.

• The velocity can be estimated dividing the length of the circumference by the time necessary for 1 revolution

• The last column (velocity2) is calculated by squaring the previous column.

Begin to spin the apparatus, maintaining a constant radius. After the spin is stabilized, have an assistant use a stopwatch to time (in seconds) 10 revolutions. Record this 10-rev time for Radius 1 in Table 2.

Shorten the length of string between the stopper and the top of the glass tube by approximately 10 cm. Pull the string through the bottom of the glass tube to shorten the distance L between the top of the glass tube and the stopper. Use the tape measure to record this new length between the top of the glass rod and the stopper as the radius for Radius 2 in Table 2.

Repeat the procedure of swinging the stopper for 10 revolutions while it is being timed. Record the time for 10 rev in Table 2 for Radius 2.

Shorten the string by another 10 cm as done before and record this new radius in Table 2 for Radius 3.

Repeat the procedure of swinging the stopper for 10 revolutions while it is being timed. Record the time for 10 rev in Table 2 for Radius  3.

Once again, shorten the string by another 10 cm. Record this new radius in Table 2 for Radius 4.

Repeat the procedure of swinging the stopper for 10 revolutions while it is being timed. Record the time in Table 2 for Radius 4.

Constant radius, variable hanging mass

Adjust the radius of the rotating mass to 0.5 m. Because this value will remain the same for this part of the experiment, we can record the length of the radius in Table 3 for all experiments in this section.

Change the number of hanging washers so that they weigh approximately 30 g. Record this hanging mass in Table 3 for Mass 1.

Use this 30-g hanging mass to perform one trial of 10 rev in a manner similar to that in Section 3.1  Record the time in Table 3 for Mass 1.

Complete the other columns for Mass 1.

Add more washers until the hanging mass is approximately equal to 40 g.  Repeat the process and complete the appropriate columns, now for Mass 2.

Repeat the experiment for a mass of 50 g  (Mass 3) and a mass for 60 g (Mass 4) as the mass of the hanging washers.

Constant radius, variable rotating mass

Adjust the radius of the rotating mass to 0.5 m. Because this value will remain the same for this part of the experiment, we can record the length of the radius in Table 3 for all experiments in this section.  Adjust the mass of the hanging washers to 50 grams.

We will increase the mass of the stopper by adding two washers each time.  To do this, untie the knot and tie two washers with the stopper. You can estimate the new rotating mass by using your data from Table 1.  If you cannot untie the knot, cut it and readjust the string to a length of 50 cm.

Repeat the previous processes and record your data and calculations for Mass 1.

Add two more washers to the stopper (total of 4 washers) and complete the data for Mass 2.

Add an additional two washers and complete the data for Mass 3.

Add two more washers and complete the data for Mass 4.

DATA CALCULATIONS

The theoretical centripetal force (FC) is given by:

Note that the hanging mass must be in kg and the resulting force will be in N.

The experimental centripetal force (FC) is given by

Complete Tables 5, 6 and 7 in which we compare the theoretical centripetal force with the measured centripetal force.

QUESTION 2

What is the relationship between the radius and the velocity of a rotating object?

QUESTION 3

What is the relationship between the velocity of rotating object and the centripetal force exerted on it?

QUESTION 4

What is the relationship between the mass of a rotating object and its velocity?

LABORATORY REPORT

Create a laboratory report  using Word or another word processing software  that contains at least these elements:

• Introduction:  what is the purpose of this laboratory experiment?

• Description of how you performed the different parts of this exercise.  At the very least, this part should contain the answers to questions 1-4 above.  You should also include procedures, etc.   Adding pictures to your lab report showing your work as needed always increases the value of the report.

• Conclusion: What area(s) you had difficulties with in the lab; what you learned in this experiment; how it applies to your coursework and any other comments.