Question

A solid sphere with a moment of inertia of I = 2/5 M R2 is rolled down an incline which is inclined at 24 degrees. The radius of the sphere is 1.6 meters. The initial velocity of the center of mass at the top of the incline is 2 m/s. As the sphere rolls without slipping down the incline, it makes 26 revolutions as it travels all the way down the incline. How long, in seconds, does it take to roll down the incline? Ignore friction

THE CORRECT ANSWER IS 12.86. PLEASE SHOW ALL THE WORK TO GET THIS ANSWER. THANK YOU!

Answer 1

DATA

SOLUTION

First, notice when they tell you "it makes 26 revolutions as it travels all the way down the incline", they are given to you the total distance traveled along the incline. When the sphere makes one revolution, this cover a distance equals to

when makes 2 revolutions

when it makes 26 revolutions

Using the Principle of conservation of energy,

isolating the final speed of the center of mass,

where is the height and is given by,

and is the moment of intertia around its center,

inserting eqs. (2) and (3) in (1),

or

Lets save this value, we are going to need it later. Now, we know that

isolating the time,

the acceleration is related to the distance by the equation,

isolating the acceleration,

inserting eq. (6) in (5),

this is equivalent to,

inserting the values given,

or