The setup of the experiment is as follows: an object of mass m is connected to a

Question

The setup of the experiment is as follows: an object of mass m is connected to a string that is wrapped around a wheel of radius R and mass M. When the mass m is released, it falls and unwraps the string from the wheel, thus the wheel rotates. You need to find the inertia of the wheel.

a) Write Newtons 2nd law for the translational motion of the hanging block. This is equation 1.

b) Write Newtons 2nd law for the rotational motion of the wheel. This is equation 2.

c) Write the equation that connects the angular acceleration of the wheel to the tangential acceleration of a point on its rim. This is equation 3.

d) Using the 3 equations, find the moment of inertia of the wheel as a function of m, R, g, and the linear acceleration of the hanging mass.

Explanation / Answer

Here ,

a) let the tension in the string is T

mg - T = m * a

b)

for the block ,

Torque = I * alpha

T * R = I * aplha

c)

alpha is the angular acceleration of rim

a is the hanging mass

alpha * R = a

d)

moment of inertia of pulley , I = 0.5 M*R^2

for the hanging string

mg - T =m * a ----1

for the pulley

T * R = 0.5 * MR^2 * a/R

T = 0.5 * M * a ----2

putting in 2

mg - 0.5 * M * a = m * a

a = m *g/(0.5 * M + m)

the acceleration of the mass is  m *g/(0.5 * M + m)

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