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Answer:
1) The hoop and a solid disc rolling without slipping down an incline plane.
Their final velocities are proportional to their moment of inertia.
The condition for moment of inertia: v = ωR
We will use conservation of energy.
<u>For the hoop:</u>
They are released from rest, so their initial kinetic energy is zero. And when they reach the bottom, their final potential energy is also zero.
The moment of inertia of a hoop is
Let's continue with the energy equations:
Similarly <u>for the solid disk</u> with a moment of inertia of (1/2)mR^2:
Comparing the final velocities, we can conclude that the solid disk reaches the bottom first.
2) The angular acceleration of the pebble is equal to the angular acceleration of the tire, since they stuck together. We can deduce the angular acceleration of the tire from the linear acceleration of the bicycle.
The kinematics equations states that
where a is the linear acceleration.
The relation with the angular and linear acceleration is
where R is the radius of the tire. Since it is not given in the question, we will leave it as R.
The angular acceleration of the small pebble is