Problem 10.57 A block of mass m = 2.9 kg is attached to a string that is wrapped

Question

Problem 10.57

A block of mass m = 2.9 kg is attached to a string that is wrapped around the circumference of a wheel of radius R = 8.2 cm . The wheel rotates freely about its axis and the string wraps around its circumference without slipping. Initially the wheel rotates with an angular speed , causing the block to rise with a linear speed v = 0.33 m/s .

Part A

Find the moment of inertia of the wheel if the block rises to a height of h = 7.0 cm before momentarily coming to rest.

Express your answer using two significant figures.

Explanation / Answer

equate initial rotational KE to final PE of the block

the initial rotational KE = 1/2 I w^2 where I is moment of inertia and w is the angular velocity

the angular velocity is related to linear velocity by w = v/r = 0.33m/s / 0.082m = 4.02rad/s

the initial rotational KE will convert to PE, so we equate

1/2 I w^2 = m g h

I = 2 m g h/w^2 = 2 * 2.9kg * 9.8m/s/s x 0.070m/(4.02rad/s)^2
I = 0.246 kg-m^2

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