1 answer

73. A hoop of radius R rotates at constant an- gular velocity 12. A small bead...

Question:

73. A hoop of radius R rotates at constant an- gular velocity 12. A small bead of mass m is attached to the hoop, with a fric

73. A hoop of radius R rotates at constant an- gular velocity 12. A small bead of mass m is attached to the hoop, with a frictional force on the bead proportional to the dif- ference in velocity between the bead and edge of the hoop, F = k(RS2 - Rw), where w is the angular velocity of the bead. If the bead begins at angular velocity wo, which of the following describes its subsequent motion? (A) w(t) = woe-kRt/m (B) w(t) = N - woe-kRt/m (C) w(t) = 12 - woe -mRt/ (D) wat) = (1 - e-kRt/m) (E) w(t) = 12 - (12 - wo)e-kRt/m

Answers

frictional force on the bead

f_c = kR(\Omega -\omega)

So,

ma = f_c

mR\dot{\omega}= kR(\Omega -\omega)

\dot{\omega} + \frac{k}{m}\omega = \frac{k}{m}\Omega

which can be solved with I.F

I.F = e^{\frac{k}{m}t}

so, the solution is

\omega e^{\frac{k}{m}t} = \Omega(e^{\frac{k}{m}}-1)+C

at t = 0 ,  

C = \omega_0

\omega = \Omega -(\Omega -\omega_0)e^{-\frac{k}{m}t}

option E

.

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