Question

(Show all the steps please)

Rotational Dynamics: Four objects of equal mass start at the top of an inclined plane. A solid cylinder, a thin walled cylinder, a solid sphere, and a thin walled sphere. All objects start at rest. Starting at rest,

A) Write down the work energy theorem including both kinetic energy terms and all potential energy terms.

B) If all four objects are released at the same moment, how long does each take to reach the bottom of the incline plane?

C) Find the ratio of translational (linear) kinetic energy to rotational kinetic energy for each object.

D) Compare the results of the race obtained in part a to the results obtained in part b.

E) If the ramp is 20 m long and inclined at an angle od 20 degrees, what is the angular speed of the solid cylinder at the bottom of the ramp.

F) What torque is required to achieve this final angular speed?

Answer 1

A) All the object are starting at rest. so , none of them have kinetic energy but only potential energy which is given as mgh where m is the mass of the respective object, g is gravity and h is the height of the ramp. Now, as they start rolling, they will undergo both rotational and translational kinetic energy. given as

1/2mv² - Translational Kinetic energy

1/2Iw² - Rotational Kinetic energy

where w = v/r (for rolling without slipping condition)

I - moment of inertia ( different for different objects)

For a solid cylinder, I = 1/2mr²

For a thin walled cylinder, I = mr²

For a solid sphere, I = 2/5mr²

For a thin walled sphere, I = 2/3mr²

I have provided you all the data...you can easily solve using this data.

for example - let's consider solid cylinder

mgh = 1/2mv² + 1/2Iw²

mgh = 1/2mv² + 1/2*1/2mr²*(v/r)²

mgh = 1/2mv² + 1/4mv²

m will cancel on both sides

gh = 1/2v² + 1/4v²

gh = 3/4v²

v = sqrt (4gh/3)

This is the velocity of solid cyliner at the bottom of the ramp

with this velocity at bottom, you can find a lot of things like time to reach the bottom

You can find the height of the ramp by using trigonometry like

sin theta = h/d

where h is height, d is length of ramp.

so, h = d*sin theta

h = 20 sin20

h = 6.84 m

Use the same method to find velocity at bottom for other objects