A heavy mass in free fall does not fall faster than a light mass. The extra weig

Question

A heavy mass in free fall does not fall faster than a light mass. The extra weight is compensated for by extra resistance. A heavy mass on a frictionless plane does not slide down faster than a light mass. A heavy ring and a light ring both roll down an inclined plane with the same speed. What about a ring and a disk? Try this with combinations of rings and disks of different masses. Describe the results and give an explanation. It may be helpful to write down an energy equation for the case of 2 rings of different radii and different masses. Also try the energy equation for a ring and a disk of the same radius and mass.

Explanation / Answer

Inertia of ring = mr^2

Inertia of disk = 1/2*mr^2

We see that for same mass and radius of ring and disk, inertia of ring is more than disk. This is becuase all the mass of the ring is located at radius r but for disk, the same mass is distributed unifrmly from radius 0 to radius r.

Torque = I = I*a/r = F*r where F is friction force

For object, mgSin - F = ma

mgSin - Ia/r^2 = ma

a = mgSin/(m + I/r^2)

When I is more, a is less.

Looking at energy: Translational energy = 1/2*mv^2

Rotational energy = 1/2*I*(v/r)^2

Hence, the more the I, the more the rotational energy and consequently lesser the translational energy since total energy remains constant.

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