How far does m travel downward between 0.410 s and 0.610 s after the motion begi

Question

How far does m travel downward between 0.410 s and 0.610 s after the motion begins?

The cylinder is changed to one with the same mass and radius, but a different moment of inertia. Starting from rest, the mass now moves a distance 0.319 m in a time of 0.450 s. Find Icm of the new cylinder.

How far does m travel downward between 0.410 s and 0.610 s after the motion begins? The cylinder is changed to one with the same mass and radius, but a different moment of inertia. Starting from rest, the mass now moves a distance 0.319 m in a time of 0.450 s. Find Icm of the new cylinder.

Explanation / Answer

M = 1.47 kg
R = 0.117 m
moment of inertia, I = 0.5MR^2 = 0.010061415 kgm^2

a. Torque, T = Ia (a = angular acc)
5.788*0.117 = Ia
a = 67.306 rad/s/s
b. Let tension in the chord be F
For the mass, mg - F = ma' [ a' is linear ac of the mass]
For the cylinder, Fr = Ia [a is angular acc]
also a' = ar
so m(g - ar)r = Ia
mgr - mar^2 = Ia
a = mgr/(mr^2 + I) = 55.84 rad/s/s
c. a' = ar = 6.533 m/s/s
S(t = 0.410) = 0.5(a')0.410^2
s(t = 0.610) = 0.5(a')0.610^2
dS = 0.5(a')(0.610^2 - 0.410^2) = 0.6664 m
d. s = 0.319 = 0.5*a'*0.45^2
a' = 3.1506 m/s/s
a = a'/r = 26.9283 rad/s/s
now, a = mgr/(mr^2 + I)
mr^2 + I = mgr/a
I = mr(g/a - r) = 0.0424692 kgm^2

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