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28 December, 09:48

What is the moment of inertia of the object starting from rest if it has a final velocity of 5.9 m/s? Express the moment of inertia as a multiple of MR2, where M is the mass of the object and R is its radius.

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  1. 28 December, 10:35
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    The moment of inertia is I = 0.126*R^2*M

    Explanation:

    We can calculate the moment of inertia of an object that starts from rest and has a final velocity using the energy conservation equation, as follows:

    Ek1 + Ep1 = Ek2 + Ep2, where

    Ek1 = kinetic energy of the object before to roll down

    Ep1 = potential energy of the object

    Ek2 = kinetic energy when the object comes down

    Ep2 = potential energy of the object at the bottom

    We have the follow:

    Ek1 = 0

    Ep1 = M*g*h

    Ek2 = ((I*w) / 2) + ((M*v^2) / 2)

    Ep2 = 0

    Replacing values:

    0 + M*g*h = ((I*w) / 2) + ((M*v^2) / 2) + 0

    where:

    M = mass of the object

    g = gravitational acceleration

    I = moment of the inertia

    w = angular velocity = v/R

    h = height

    M*g*h = ((1/2) * I * (v^2/R^2)) + ((M*v^2) / 2)

    M*9.8*2 = (I * (5.9^2) / (2*R^2)) + ((5.9^2 * M) / 2)

    19.6 * M = ((17.4*I) / R^2) + 17.4*M

    Clearing I, we have:

    I = 0.126*R^2*M
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