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A disc performs angular (torsional) oscillations using a wire with torsional constant k. Its initial angular amplitude is theta0. At the instant when the disc passes through angular displacement theta0/2 (half the amplitude), an impulsive torque is applied that instantaneously increases its angular velocity to 1.5 times its value at that position. Find the new angular amplitude of oscillation after the impulse.

1. (theta0/4)*sqrt(31)
2. (theta0/2)*sqrt(3)
3. 3*theta0/4
4. (theta0/4)*sqrt(41)

Correct answer: (theta0/4)*sqrt(31)

Solution

Boosting the angular velocity by factor 1.5 multiplies the kinetic energy by 2.25. The new total energy (31/32)*k*theta0² corresponds to a new amplitude theta_new satisfying (1/2)*k*theta_new² = (31/32)*k*theta0², giving theta_new = (theta0/4)*sqrt(31).

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