A wheel of moment of inertia of 5.00 kg m2 starts from rest and accelerates under a constant torque of 3.00 N m for 8.00 s. What is the wheel's rotational kinetic energy at the end of 8.00 s?

Question

BlackTom AI · Accepted Answer

Torque produces angular acceleration: α = τ / I. Here τ = 3.00 N·m and I = 5.00 kg·m^2, so α = 3.00 / 5.00 = 0.600 rad/s^2.

1) Relevant knowledge points and concepts
- Newton’s second law for rotation: τ = I α.
- Angular kinematics for constant α: ω = ω0 + α t.
- Rotational kinetic energy: K = (1/2) I ω^2.
- Work-energy for rotation (alternate): work by torque = τ Δθ = change in rotational kinetic energy.

2) Complete reasoning process step by step
- Compute angular acceleration α from τ and I.
- Find angular speed ω after t = 8.00 s starting from rest (ω0 = 0).
- Compute rotational kinetic energy using K = (1/2) I ω^2.
(Also show an alternative using work = τ Δθ to confirm the same K.)

3) Calculations with intermediate steps
- α = τ / I = 3.00 N·m / 5.00 kg·m^2 = 0.600 rad/s^2.
- ω = ω0 + α t = 0 + (0.600 rad/s^2)(8.00 s) = 4.80 rad/s.
- K = (1/2) I ω^2 = 0.5 × 5.00 kg·m^2 × (4.80 rad/s)^2
  = 2.5 × (4.80^2) = 2.5 × 23.04 = 57.60 J.
Alternative check via work:
- Angular displacement Δθ = 0.5 α t^2 = 0.5 × 0.600 × (8.00)^2 = 0.300 × 64 = 19.2 rad.
- Work = τ Δθ = 3.00 × 19.2 = 57.6 J = K (consistent).

4) Why the correct answer is correct
Because a constant torque produces a constant angular acceleration; using the resulting angular speed in the rotational kinetic energy formula gives 57.6 J. The alternative work approach yields the same value, confirming consistency.

5) Why the other options are incorrect
- 64.0 J: would result from an incorrect ω (e.g., using ω = 5.0 rad/s) or arithmetic error.
- 78.8 J and 122 J: both result from incorrect use of formulas or wrong numerical substitution (e.g., wrong α, wrong squaring, or forgetting the 1/2 factor).

6) Conclusion
The rotational kinetic energy after 8.00 s is 57.6 J, which matches the provided correct answer.

A wheel of moment of inertia of 5.00 kg m2 starts from rest and accelerates under a constant torque of 3.00 N m for 8.00 s. What is the wheel's rotational kinetic energy at the end of 8.00 s?单项选择题

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