An oversized yo-yo is made from two identical solid disks each of mass M=2.00kg

Question

An oversized yo-yo is made from two identical solid disks each of mass M=2.00kg and radius R=10.0 cm. The two disks are joined by a solid bylinder of radius r=4 cm and mass m=1 kg. Take the center of the bylinder as the axis of the system, with positive torques directed to the left along this axis. Light string is wrapped around the cylinder, and the system is then allowed to drop from rest.
a. what is the moment of inertia? Give a symbolic answer b. what torque does gravity exert on the system with respect to the fiven axis? c. Take downward as negative coordinate direction. Is torque exerted by the tension positive or negative? what about the translational acceleration? d. What is an equation for the angular acceleration in terms of translational acceleration a and radius r. e. Write Newton's second law for the system in terms of m, M, a, T, and g. F. Write Newton's second law for rotation in terms of I, angular acceleration, T and r. G. Elimintate rotational second law found i part d and find a symbolic expression for the accelerationa in terms of m, M, g, r and R. h. What is the numerica value for the system's acceleration? I. What is the tension in the string? J. How long does it take the system to drop 1.oom from rest?

Explanation / Answer

R = 0.1 m
r = 0.04 m

First find the moment of inertia of the yo-yo:
I = I = 2 x (1/2)MR2 + (1/2)mr2 =MR2 + mr2

= r F
The force is just the weight of the yo-yo
= r(M + m)

Newton's second law for rotating bodies
= I
r(M + m) = (MR2 + mr2)
Solve for .

a = r will give the translational acceleration

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