A cylinder with a moment of inertia (about its axis of symmetry), mass m, and ra

Question

A cylinder with a moment of inertia (about its axis of symmetry), mass m, and radius r has a massless string wrapped around it which is tied to the ceiling (Figure 1) . At time the cylinder is released from rest at height above the ground. Use for the magnitude of the acceleration of gravity. Assume that the string does not slip on the cylinder. Let v represent the instantaneous velocity of the center of mass of the cylinder, and let w represent the instantaneous angular velocity of the cylinder about its center of mass. Note that there are no horizontal forces present, so for this problem v=-vy and w=-wz.

The string constrains the rotational and translational motion of the falling cylinder, given that it doesn't slip.

A) What is the relationship between the magnitude of the angular velocity and that of the velocity of the center of mass of the cylinder? *express w in terms of v & r*

B) Using Newton's 2nd law, complete the equation of motion in the vertical direction that describes the translational motion of the cylinder. Epress in terms of Tension T of vertical string, g & m PLease show work on how to derive answer

Explanation / Answer

ma_y = T - mg except that they don't want T to appear in your answer. Therefore we need to find an expression for T in terms of g, m, r, and I. We can get it from I*alpha = -Tr Tr = -I*alpha T = -I*alpha/r But they also don't want alpha to appear in the answer. From Omega = v/r you can derive alpha = a_y/r T = -I*alpha/r T = -I*(a_y/r)/r = -I*a_y/r^2 ma_y = T - mg ma_y = -I*a_y/r^2 - mg ma_y + I*a_y/r^2 = -mg Distributive law a_y(m + I/r^2) = -mg a_y = -mg / (m + I/r^2)

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