A coin and a ring have the same mass and radius. Starting from rest and the same

Question

A coin and a ring have the same mass and radius. Starting from rest and the same height, they roll down the same incline. In other words, they both start with the same gravitational potential energy. Which one of the following statements is true about the coin and ring when they reach the bottom of the incline?

C. The increase in the coin's total kinetic energy is greater than the increase in the ring's total kinetic energy.

C. The speed of the coin's center of mass is greater than the speed of the ring's center of mass.

A. The increase in the coin's total kinetic energy is less than the increase in the ring's total kinetic energy. B. The increase in the coin's total kinetic energy is equal to the increase in the ring's total kinetic energy.

C. The increase in the coin's total kinetic energy is greater than the increase in the ring's total kinetic energy.

A. Coin’s Krotational = ring's Krotational,and coin's KCM = ring's KCM B. Coin’s Krotational = ring's Krotational,and coin's KCM < ring's KCM C. Coin’s Krotational = ring's Krotational,and coin's KCM > ring's KCM D. Coin’s Krotational < ring's Krotational,and coin's KCM < ring's KCM E. Coin’s Krotational < ring's Krotational,and coin's KCM = ring's KCM F. Coin’s Krotational < ring's Krotational,and coin's KCM > ring's KCM G. Coin’s Krotational > ring's Krotational,and coin's KCM > ring's KCM H. Coin’s Krotational > ring's Krotational,and coin's KCM = ring's KCM I. Coin’s Krotational > ring's Krotational,and coin's KCM < ring's KCM A. The speed of the coin's center of mass is less than the speed of the ring's center of mass. B. The speed of the coin's center of mass is equal to the speed of the ring's center of mass.

C. The speed of the coin's center of mass is greater than the speed of the ring's center of mass.

Explanation / Answer

Coin is a disc, so its moment of Inertia I = mr2/2

I of ring = mr2

Now, by conservation of mechanical energy,

Decrease in Potential energy (U)= Rotational Kinetic Energy(Krot) + Translational Kinetic Energy (Ktra)

Krot = 1/2*I*w2 where w is the angular velocity

Ktra = 1/2 mv2 where v is the velocity of the centre of mass

Applying conservation of energy for disc

mgh = 1/2 * mr2/2 * w2 + 1/2 * m * v2

also, for rolling v = wr, so substituting w = v/r in the above equation

mgh = 1/2 * mr2/2 * (v/r)2 + 1/2 * m * v2

Solving we get v = (4gh/3)1/2

Ktra = 1/2 * m * v2 = 2mgh/3

Krot = 1/2 * mr2/2 * (v/r)2 = mgh/3

Applying conservation of energy for ring

mgh = 1/2 * mr2 * w2 + 1/2 * m * v2

also, for rolling v = wr, so substituting w = v/r in the above equation

mgh = 1/2 * mr2 * (v/r)2 + 1/2 * m * v2

Solving we get v = (gh)1/2

Ktra = 1/2 * m * v2 = mgh/2

Krot = 1/2 * mr2 * (v/r)2 = mgh/2

Answers

By conservation of energy potential energy is converted to total kinetic energy (roational + translational). As same amout of potential energy is lost same amout of kinetic energy is gained by both the bodies. So option B

From above calculations, Krot of coin< Krot of ring and Ktra of coin > Ktra of ring, so option F

v of coin is (4gh/3)1/2 and v of ring is (gh)1/2 so option C

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