After class one day, two physics students get into an argument about rotational

Question

After class one day, two physics students get into an argument about rotational dynamics. Jack thinks the nickel in his pocket would beat s silver in a race down an incline, but Jill is convinced her ring would win. To settle their argument, Jack and Jill climb up a 20 m hill inclined at 30 degree to the horizontal. Both objects are released from rest at the top of the hill and roll without slipping all the way to the bottom (of course, Jack and Jill come tumbling after). What is the translational speed of each object when it reaches the bottom? How long does each object take to get from the release point to the bottom of the hill?

Explanation / Answer

For the Nickel

Gravitational Poteteial Energy = Linear Kinetic Energy

mgh = 0.5*m*v^2

gh = 0.5*v^2

9.8*20 = 0.5*v^2

v = 19.789 m/sec

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For Ring

Gravitational POtetneial Energy = Rotational Kinetic Energy + Linear Kinetic Energy

mgh = 0.5*I*w^2 + 0.5*m*v^2

where I is MOment of inertia

W is angualr velocity

m is mass

As For Ring I = mr^2 and the relation betweeb V,r W is w = v/r

thus mgh = 0.5*m*v^2 + 0.5*m*v^2

mgh = mv^2

v = sqrt(gr)

to get V, use the radius of the ring

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