A solid rigid object with a moment of inertia of 12 kgm2 has an initial angular velocity of 9 rads-1 clockwise. A torque of 44 Nm was then applied to the object in the opposite direction to the initial angular velocity. After 3 s the object will have an angular speed closest to:

Question

BlackTom AI · Accepted Answer

We start by identifying the given quantities and the effect of the applied torque. The object has a moment of inertia I = 12 kg·m^2, an initial angular velocity ω_i = 9 rad/s (clockwise), and a torque τ = 44 N·m applied opposite to the initial rotation for a duration t = 3 s. Since the torque is opposite to the initial rotation, it will produce angular deceleration.

Option A: 2 rads-1
First, compute the angular deceleration: α = τ / I = 44 / 12 = 11/3 ≈ 3.666... rad/s^2, directed opposite to the initial rotation. Over 3 seconds, the change in angular velocity is Δω = α t ≈ (11/3) × 3 = 11 rad/s. Starting from ω_i = 9 rad/s, the final angular velocity is ω_f = ω_i − Δω = 9 − 11 = −2 rad/s, i.e., 2 rad/s in the opposite (counterclockwise) direction. The magnitude is 2 rad/s, which matches option A.

Option B: 4 rads-1
If the final speed were 4 rad/s, the net change in angular velocity would be Δω = ω_i − ω_f = 9 − 4 = 5 rad/s. This would require a deceleration of 5 rad/s over 3 s, i.e., α = Δω / t ≈ 5/3 ≈ 1.667 rad/s^2, which does not equal the calculated α = 11/3 rad/s^2. Therefore this value is inconsistent with the given torque and time.

Option C: 3 rads-1
A final speed of 3 rad/s implies Δω = 9 − 3 = 6 rad/s (deceleration of 6 rad/s over 3 s gives α = 2 rad/s^2). This α is not equal to the required α = 11/3 rad/s^2 from τ/I, so this option does not fit the data.

Option D: 1 rads-1
A final speed of 1 rad/s would require a Δω = 8 rad/s over 3 s, giving α ≈ 8/3 ≈ 2.667 rad/s^2, still not matching the required α = 11/3 rad/s^2. Thus this option is also inconsistent with the given torque and duration.

A solid rigid object with a moment of inertia of 12 kgm2 has an initial angular velocity of 9 rads-1 clockwise. A torque of 44 Nm was then applied to the object in the opposite direction to the initial angular velocity. After 3 s the object will have an angular speed closest to:

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