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# Motion on a circular path. (horizontal and vertical)

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1. The tub of a washer goes into its spin-dry cycle, starting from rest and reaching an angular speed of 3.0 rev/s in 7.0 s. At this point the person doing the laundry opens the lid, and a safety switch turns off the washer. The tub slows to rest in 10.0 s. Through how many revolutions does the tub turn during this 17 s interval? Assume constant angular acceleration while it is starting and stopping.

2. An engineer wishes to design a curved exit ramp for a toll road in such a way that a car will not have to rely on friction to round the curve without skidding. She does so by banking the road in such a way that the force of the centripetal acceleration will be supplied by the component of the normal force toward the center of the circular path.
(a) Show that for a given speed of v and a radius of r, the curve must be banked at the angle such that tan = v2/rg. (Do this on paper. Your instructor may ask you to turn in this proof.)
(b) Find the angle at which the curve should be banked if a typical car rounds it at a 54.0 m radius and a speed of 14.0 m/s.

3. A 0.420 kg pendulum bob passes through the lowest part of its path at a speed of 3.20 m/s.
(a) What is the tension in the pendulum cable at this point if the pendulum is 80.0 cm long?
(b) When the pendulum reaches its highest point, what angle does the cable make with the vertical?
(c) What is the tension in the pendulum cable when the pendulum reaches its highest point?

4. A roller-coaster vehicle has a mass of 492 kg when fully loaded with passengers (Fig. P7.28).

(a) If the vehicle has a speed of 18.0 m/s at point A, what is the force of the track on the vehicle at this point?
(b) What is the maximum speed the vehicle can have at point B in order for gravity to hold it on the track?

5. An air puck of mass 0.25 kg is tied to a string and allowed to revolve in a circle of radius 1.0 m on a frictionless horizontal table. The other end of the string passes through a hole in the center of the table, and a mass of 1.2 kg is tied to it (Fig. P7.25). The suspended mass remains in equilibrium while the puck on the tabletop revolves.
(a) What is the tension in the string?
(b) What is the force causing the centripetal acceleration on the puck?
(c) What is the speed of the puck?

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#### Solution Preview

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1. The tub of a washer goes into its spin-dry cycle, starting from rest and reaching an angular speed of 3.0 rev/s in 7.0 s. At this point the person doing the laundry opens the lid, and a safety switch turns off the washer. The tub slows to rest in 10.0 s. Through how many revolutions does the tub turn during this 17 s interval? Assume constant angular acceleration while it is starting and stopping.

In analogy to the equations of motion for motion in one dimension the equations for the circular or rotatory motion are given as

where 0 is the initial angular velocity,  is the angular velocity at time t,  is angular acceleration and  is the angular turned.

(a) Angular acceleration in the first 7 second

3.0 = 0 + *7.0 gives  = 3/7 rev/s2.

Substituting in the third equation we have

9.0 = 0 + 2*(3/7)* gives 1 = 21/2 = 10.5 revolutions.

(b) Angular acceleration in the last 10 second

0 = 3.0 + *10.0 gives  = - 3/10 rev/s2.

Substituting in the third equation we have

0 = 9 + 2*(- 3/10)* gives 2 = 15 revolutions.

Hence the angle turned in 17 second 1 + 2 = 10.5 + 15 = 25.5 rev.

25.5 rev

2. An engineer wishes to design a curved exit ramp for a ...

#### Solution Summary

There are solutions of five problems related to motion on a circular path. Motion on both horizontal and vertical circles is discussed. The problems are on motion in horizontal and vertical circle, centripetal force, banking on curves, centrifugal reaction etc.

The solutions will help you learn all about circular motion.

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