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A smooth straight slot lies along a diameter of a disc of radius R that rotates counter-clockwise about its vertical central axis with constant angular speed omega (omega-vector = omega k-hat). With the centre as origin and x-axis along the slot, a small block of mass m is gently placed in the slot at r = (R/2) i-hat at t = 0 and is constrained to move only along the slot. Using F_rot = F_in + 2m(v_rot x omega) + m(omega x r) x omega, find the radial position r(t) of the block along the slot.

1. R/4 (e^(2*omega*t) + e^(-2*omega*t))
2. R/2 cos(2*omega*t)
3. R/2 cos(omega*t)
4. R/4 (e^(omega*t) + e^(-omega*t))

Correct answer: R/4 (e^(omega*t) + e^(-omega*t))

Solution

The radial equation x'' = omega² x with x(0) = R/2, x'(0) = 0 gives x = (R/2) cosh(omega t) = R/4 (e^(omega t) + e^(-omega t)).

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