The disk wheel has a radius of 0.200 m and a mass of 3.00 kg, which is uniformly

Question

The disk wheel has a radius of 0.200 m and a mass of 3.00 kg, which is uniformly distributed throughout its volume. It is released from rest at a height of 1.35 m from the bottom of the incline. Calculate its translational and rotational speed at the bottom of the incline. The disk wheel is now released from rest from the same point along side a hoop. The hoop may or may not have the same radius and mass as the disk wheel. We may conclude that the disk wheel will arrive at the bottom the hoop. A second solid disk wheel is now raced against the first disk wheel. It is observed that the first disk wheel arrives at the bottom earlier than the second disk wheel. We may conclude that The radius of the second disk wheel is larger than the radius of the first disk wheel. The mass of the second disk wheel is not uniformly distributed throughout its volume, but instead is concentrated near the edges of the disk wheel. The mass of the second disk wheel is larger than the radius of the first disk wheel. The radius of the second disk wheel is smaller than the radius of the first disk wheel. None of the above can be concluded from this information. This situation is impossible. The mass of the second disk wheel is smaller than the mass of the first disk wheel. The mass of the second disk wheel is not uniformly distributed throughout its volume, but instead is concentrated near the center of the disk wheel.

Explanation / Answer

Given that

The mass of disck m = 3.0 kg

Radisu R = 0.2 m

The moment of inertia of disck I = 0.5 m R2

= 0.5 x 3.0 x 0.22

= 0.06 kg-m2

When the disc rolls without slipping

v = R w

Apply conservation of energy

Ei = Ef

m g h = 0.5 mv2 + 0.5 Iw2

m g h = 0.5 mv2 + 0.5 I(v/ R)2

v = [ 4 g h / 3]1/2 =   [ 4 * 9.8 * 1.35 / 3]1/2 = 4.2 m/s

This is the linear speed at the bottom of the incline plane.

The angular speed at the bottom of the inclined plane is

w = v / R = 4.2 / 0.2 = 21 rad/s

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Which one comes first is depends on the smaller I / MR2 value

If disc two has larger radius then it reaches the bottom first.

Hence correct option is

The radius of second disk wheel is larger than the first disk wheel.

  

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