What is uniform circular motion give two examples which force is responsible for that

What is uniform circular motion give two examples which force is responsible for that
A carousel spins with uniform circular motion.
(Credit: User Rept0n1x, via Wikimedia Commons)

Newton's First Law of motion states that an object moving at constant speed will continue that motion unless acted on by an outside force. This means that circular motion can only happen if there is a "center seeking" force – otherwise things would just travel in a straight line, rather than the curved line of a circule. Centripetal means 'center seeking', so centripetal force is used to refer to the force experienced by an object traveling in a circle. For example, when someone spins a ball attached to a rope horizontally above his head, the rope transmits a centripetal force from the muscles of the hand and arm, causing the ball to move in a circular path.

Centripetal forces cause centripetal accelerations. In the special case of the Earth's circular motion around the Sun – or any satellite's circular motion around any celestial body – the centripetal force causing the motion is the result of the gravitational attraction between them.

What is uniform circular motion give two examples which force is responsible for that
Animation of uniform circular motion.

What is uniform circular motion give two examples which force is responsible for that
Close up of the satellite showing velocity and acceleration vectors.

The arrows (or vectors) show the direction of the circular velocity (v, always tangent to the circular path) and the circular acceleration (a) caused by a centripetal force. Centripetal means center-seeking. Centripetal forces are always directed toward the center of the circular path.

By definition, acceleration is the rate of change in velocity of an object, and velocity is determined by dividing the distance travelled by the time interval it took to cover that distance. In the special case of circular motion, the distance covered is the circumference of a circle or 2πr, where π is the mathematical constant and r is the radius fo the circle. The time interval for an object to travel once around its circular path is called the period and is represented by T.

What is uniform circular motion give two examples which force is responsible for that

The equation for centripetal acceleration is:

What is uniform circular motion give two examples which force is responsible for that

Putting those together, we get the equation for centripetal force:

What is uniform circular motion give two examples which force is responsible for that

What is uniform circular motion give two examples which force is responsible for that
Take a quiz on uniform circular motion

What is uniform circular motion give two examples which force is responsible for that
Return to solving this using Newton's law of universal gravitation

What is uniform circular motion give two examples which force is responsible for that
Return to the beginning and try another approach

If you have ever been on an amusement park ride that travels in a curved or circular path, then you have experienced a force, called a centripetal force, pushing you into the ride. Whether it's the back wall of the "Roundup" or "Rotor", the ride where the floor drops from beneath your feet, or the seat belt of the "roller coaster" that supplies the force, you are constantly being accelerated toward the ride's center of curvature. If—what you fear most on such a ride—this force were suddenly removed, you would move off in a direction tangent to the circular path. That's what happens when you go over a hill on the roller coaster just before the seat belt takes effect. Figure 1 compares the motion in the presence of a centripetal force to the resulting motion of a body if the centripetal force were to cease suddenly.

What is uniform circular motion give two examples which force is responsible for that

Figure 1: Object in circular motion

As another example consider a ball, attached to a string and whirled in a circle as shown in Fig. 2a. The tension in the string applies the centripetal force to the ball, causing it to move in a circular path. The string pulls the ball toward the center of the circle while the ball pulls outward on the string and hence on your hand in accordance with Newton's third law of action and reaction. So this outward force does not act on the ball, though it is commonly and incorrectly referred to as the centrifugal force acting on the ball. When the centripetal force is discontinued, such as in Fig. 2b when the string breaks, the object moves in the direction of the velocity at that instant. This direction is tangential to the circle at that point.

What is uniform circular motion give two examples which force is responsible for that

Figure 2: Top view of a ball on a string before and after the string breaks

The centripetal force that holds you in the ride can be determined with a few measurements and calculations. In this experiment you will determine what variables must be known to determine the centripetal force required to keep a mass moving in a circular path with a constant speed.

Newton's second law states that the net force exerted on a moving body is equal to the product of a body's mass and its acceleration

( 1 )

F

What is uniform circular motion give two examples which force is responsible for that
net = ma
What is uniform circular motion give two examples which force is responsible for that

When applying Newton's second law to circular motion, it is convenient to use coordinates that are parallel and perpendicular to the object's motion. For a body moving in a straight line, the acceleration is due to a change in the magnitude of the velocity. For a body moving in a circular path with constant speed the magnitude

| v

What is uniform circular motion give two examples which force is responsible for that
|

of the velocity does not change, but the direction of the velocity vector

v

What is uniform circular motion give two examples which force is responsible for that

constantly changes. In this case the object has no acceleration in the direction parallel to its motion and in Eq. (2) is zero. This motion is called uniform circular motion—motion in a circular path at constant speed. Since the velocity vector is changing in time, the object in uniform circular motion is accelerating. Conceptually, using parallel and perpendicular coordinates is convenient because the parallel force is responsible for changes in speed and the perpendicular force (or centripetal force) is responsible for changes in direction.

What acceleration is needed to keep the body moving in a circle with constant speed? Refer to Fig. 3a below. At a certain instant the particle is located at the tip of the radius vector

r

What is uniform circular motion give two examples which force is responsible for that

. At another instant later the particle is located at the tip of the radius vector

r

What is uniform circular motion give two examples which force is responsible for that
'

. In the small time interval the body moves along an arc of length . If is small enough, the arc length is approximately equal to the distance between the position of the body at the beginning of the interval and the position at the end of the interval. This straight-line segment, together with the two radii, forms the isosceles triangle shown in Fig. 3a.

What is uniform circular motion give two examples which force is responsible for that

Figure 3: Geometrical considerations

Figure 3b shows the velocity vectors at the two positions. Notice that the magnitude of the velocity vector is not changing. In Fig. 3c the two velocity vectors are redrawn, without changing their lengths or orientations. The arrow joining the tips of the velocity vectors represents the change in the velocity of the particle due to the change in direction during the time interval . Note that as gets smaller, the angle between the direction of and both and comes close to a right angle 90°.

Since the tangent to a circle is perpendicular to its radius, it follows that the angle between and as shown in Fig. 3c is the same as the angle between the two radii in Fig. 3a. From trigonometry we know that these two triangles are congruent, and therefore the ratios of their sides must be equal. So we can write the following expression Dividing both sides by gives Of course is the acceleration of the body, and it is in the direction of . From Fig. 3c we can see that the acceleration is directed toward the center of the circle. The magnitude of the centripetal acceleration is given by and the centripetal force is

Since it is difficult to measure the velocity of the body directly, you will instead compute the velocity from quantities that are easier to measure. The magnitude of the velocity vector can be determined by measuring the distance that the object travels per unit time. If is the period (length of time needed for the object to make one complete revolution), then the speed is equal to the distance traveled in that one revolution divided by the period. Substituting this into Eq. (6) for the centripetal acceleration gives

( 9 )

ac = = 4π 2f 2r = 4π 2T −2r

where is the number of revolutions per second measured in Hertz. Now Eq. (7) can be written in terms of the measurable quantities , , and as

In this experiment you will measure the period of rotation for an object and calculate the centripetal force acting on it. You will compare this centripetal force with an equivalent force needed to maintain the object at the same radius.

  • Circular motion apparatus
  • Assorted weights
  • Bubble level
  • Balance
  • Stop watch
  • Meter stick
  • Spring

What is uniform circular motion give two examples which force is responsible for that

While it might be more fun to carry out this study on a midway ride, a simple laboratory apparatus shown in Fig. 4 below will be used to examine the nature of centripetal force. By using the centripetal force apparatus you can measure the frequency of rotation of an object moving in a circular path of a known radius. Eq. (10) can then be used to compute the centripetal force acting on the object.

Description of apparatus

Figure 4 shows the apparatus and its various components. The bottom of the base is provided with adjustable screws that can be used to level the entire apparatus. The crossbar with the counter-weight at its end can be moved to change the position of the rotating mass. The pointer can be moved along the slot and positioned just under the tip of the rotating mass. This helps to measure the radius of the circular path of the rotating mass.

What is uniform circular motion give two examples which force is responsible for that

Figure 4: Sketch showing the components of the apparatus

In this apparatus the centripetal force is provided by the spring. When the spring is attached to the rotating mass , it gets pulled in as shown in Fig. 5a and Fig. 6b. To bring the rotating mass back above the pointer or radius indicator, the string is passed over the pulley and enough mass added at the end of the string to bring the rotating mass back above the radius indicator.

What is uniform circular motion give two examples which force is responsible for that

Figure 5: Sketch showing initial set-up

What is uniform circular motion give two examples which force is responsible for that

Figure 6: Photo of initial set-up

In Fig. 5b and Fig. 6c the weight of the hanging mass just balances the elastic force of the spring when the rotating object rests above the pointer or radius indicator post. In Fig. 7 the spring is stretched by the same amount as the mass rotates in a path with the same radius as that set in Fig. 5b. The magnitude of the two forces and should be in close agreement since both forces produce the same elongation of the spring.

What is uniform circular motion give two examples which force is responsible for that

Figure 7: Sketch showing the circular path of the object

Procedure A: Measuring the period of rotation

1

Level the apparatus with the three adjusting screws in the base. The apparatus is level when the crossbar remains fixed in any position. If necessary, use a bubble level to assist you in this process.

2

Weigh the rotating mass and enter this value on the worksheet.

3

Set the radius indicator for the smallest radius and measure . The radius is measured from the center of the post (axis of rotation) to the center of the radius indicator. Record in the Data Table on the worksheet.

4

Move the crossbar until the tip of the rotating mass is above the radius indicator.

5

Attach the spring to the rotating mass. Attach the string to the rotating mass, pass the string over the pulley, and attach it to the mass hanger at the other end. Add masses to the hanger until the tip of the rotating mass is above the radius indicator. Record the value of the total hanging mass in Data Table 1 on the worksheet.

6

Remove masses and hanger and rotate the crossbar. Make sure the tip of the rotating mass hits the radius indicator every revolution. You may have to practice a little before you start taking data. It is important to spin the shaft uniformly. CAUTION: Make sure to keep hair and clothing away from the apparatus when it is in motion! Using a timer record the time required for 50 revolutions. Record this time in Data Table 1 on the worksheet.

7

Unhook the spring and move the radius indicator out by about 1 cm. Measure and record this new radius value in Data Table 1.

8

Repeat steps 4 through 6 and enter the values in Data Table 1.

9

Repeat step 7 for three more positions of the radius indicator, for a total of 5 radii.

Checkpoint 1:
Ask your TA to check your Data Table 1 values before proceeding.

10

Calculate and record the period of rotation in Data Table 2.

11

Calculate and record the frequency of rotation in Data Table 2.

12

Calculate and record the centripetal acceleration in Data Table 2.

13

Calculate the centripetal force.

14

Calculate the percent difference between the experimental value of the centripetal force and the force of the hanging mass, and record the values in Data Table 2. See Appendix B. Note: If your percent difference is greater than 15% for one trial, you must redo that trial.

Checkpoint 2:
Ask your TA to check your Data Table 2 values and calculations.

Procedure B: Plot of versus

15

Using Excel, graph versus . See Appendix G.

16

Use the trendline option in Excel to find the slope. See Appendix H.

17

From the slope determine the value of the rotating mass.

18

Compare the value of the rotating mass obtained from the slope with the measured value by computing the percent difference.

Checkpoint 3:
Ask your TA to check your graph and calculations.