Centripetal Acceleration and Centripetal Force
To experimentally verify the formula for centripetal acceleration and force and learn how to determine the linear speed of an object from its angular speed as it performs uniform circular motion.
A computer with internet connection, a calculator (The built-in calculator of the computer may be used.), paper, and pencil
An object performing uniform circular motion is constantly under the action of a force that acts toward its center of rotation and has a magnitude of
where m is the mass of the object, v its liner speed tangent to the circular path, and r its radius of rotation.
An object tends to move along a straight line if it is under the action of a force that has a constant direction, but if the direction of the force keeps changing such that it is always directed toward the center of a circle, the object is then forced to follow a circular path. Fig. 1 shows a small object of mass m that is connected to a string of length r and is being spun in a horizontal plane at constant linear speed, v. Note that the velocity is not constant. This is because of the fact that the direction of velocity keeps changing in circular motion. The velocity vector v is always perpendicular to the radius of rotation r.
If the radius of rotation ,r, is very big, the object is not greatly forced to curve, and therefore, a small centripetal force is needed for its mild curved motion. You will notice this in the following applet from the way the necessary hanging mass becomes smaller when greater radius of rotation is selected.
Click on the following link:
The red mass m attached to a cord of length r = 10.0m (as shown in the top left corner) is performing uniform circular motion. The length of the cord can be changed by moving the hanging mass M up or down by the mouse. As you move the hanging mass up and down, the radius of rotation r changes as its value can be read on the top left corner. r may be set at any desired value. For simplicity of calculations, we will assume that the mass of the rotating mass is m=1.00kg.
The accepted value for centripetal acceleration can be read from the space at the top where it says Mg/m. Note that this ratio has units of acceleration, or simply m/s2 . Setting m = 1.00kg , the product Mg gives us the centripetal acceleration of the rotating mass m.
The measured value for centripetal acceleration can be found in a different way. Counting the number of turns in each case and read the corresponding elapsed time. Since the length traveled on the circle in each turn is 2pr, the length traveled in n turns is Dx = 2prn. The elapsed time is t or Dt. The average speed is therefore, v = Dx/ Dt as shown in the following Table as well. Now, knowing v, the centripetal acceleration ac can be calculated as v2/r.
Accordingly, the measured and accepted values for centripetal force can also be found as shown in the Table.
Note: When you clean any previous run and reset the applet, it is ready to go. Holding down on a left click makes the applet running and releasing it stops it. You need to practice a few times to become comfortable with it.
1) Let the initial data be: r = 15.0m, and w = 0.69rd/s, and as was mentioned m = 1.00kg.
2) click on the page and put m in motion. Let it run for 5 turns, for example, and at the end of the 5th turn release the mouse and it will stop. Record the values of t and n, the number of turns, in the Table below.
3) Calculate V, V2/r , and read Mg/m and record them in the Table.
4) Calculate the related %error and record it.
5) Repeat the experiment for cases 2 through 6 by just changing the radius of rotation as shown in the Table. Of course, the hanging mass M changes to adjust for the necessary centripetal force. You need more number of turns as mass m turns faster for smaller radii. This will make the error smaller because of your reaction time and your judgment at start or stop instants.
Note: If the applet starts acting strange, refresh the screen.
Given: m = 1.00kg, g = 9.8 m/s2.
Comparison of the results:
Write down the percent error formula used.
State your conclusions of the experiment.
Provide a discussion if necessary.
Give two examples on objects executing circular motion and determine the source of centripetal force for each.
Suppose you tie a rock to a string and spin it in a vertical plane. In what direction will the rock move if the string breaks exactly when the rock has passed the lowest point and is in a horizontal position?
Using V = 2πrf, derive a formula for Fc in terms of m, r, and frequency, f. Simplify the formula after you substitute for v in the centripetal force formula. Note that f is the frequency of rotation or the number of turns per second.