Consider a ring, a cylinder, and a ball with equal radius and equal mass. The ob

Question

Consider a ring, a cylinder, and a ball with equal radius and equal mass. The objects are at rest at the top of an inclined plane as shown. There is friction and the object roll without slipping.

1) Which object reaches the bottom first? Explain.

2) What would happen if the masses of the objects were doubled?

3) Do the objects have the same linear velocity when they reach the bottom of the ramp? Explain.

4) How would your answer change if there was no friction?

5) What percentage of energy is lost to friction for the cylinder?

Explanation / Answer

Before answering, we need to realize that there is an underlying assumption that all the objects have the same mass. Consequently, the bodies have different densities.

Given that, lets start our analysis. For any process, energy is conserved. For this case, there are 3 different types of energy: Translational kinetic energy (energy because it is moving forward), Rotational kinetic energy (due rotation) and potential energy. More formally, we can write as:

E=mgh+mv2/2+I2/2E=mgh+mv2/2+I2/2

Another assumption made here is that the objects are rolling without slipping. Mathematically, this means:

v=rv=r

If this assumption is accounted for, our original equation becomes:

E=mgh+mv2/2+Iv^2/2r^2

Now lets analyze. For all the bodies m, h, r and g are the same. Difference lies in I and it is inversly proportional to the velocity (or rotation).

I (moment of inertia ) of ball is least (2/5 MR^2). So velocity will be most, So ball will reach first.

2) if masses will be doubled, still the ball will reach first

3) No velocity is different as found in first part

4) if there is no friction, all will reach at same time and will have same linear velocity on reaching.

5) zero energy is lost in friction, as point of contact doesnt move in pure rolling

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