Difficulty: Easy
Correct Answer: I α
Explanation:
Introduction / Context:The fundamental relation between torque and angular acceleration in rotational dynamics mirrors Newton’s second law for translation. Correctly applying tau = I * alpha is essential in analyzing gears, flywheels, rotors, and any accelerating rotating machinery.
Given Data / Assumptions:
Concept / Approach:Analogous to F = m * a, rotational motion follows tau = I * alpha, where tau is the net torque about the axis. Dimensional consistency: N·m on LHS; on RHS, (kg·m^2) * (rad/s^2). Since rad is dimensionless, kg·m^2/s^2 equals N·m, confirming the relation.
Step-by-Step Solution:
Start from rotational form of Newton’s second law: sum(tau) = I * alpha.Identify I and alpha from the problem statement.Conclude tau = I * alpha (no squares, no inverses).Verification / Alternative check:
Check units: I (kg·m^2) * alpha (1/s^2) = kg·m^2/s^2 = N·m → matches torque.Why Other Options Are Wrong:
I2 α and I2/α: squaring I is dimensionally incorrect for torque.I/α: inversion produces dimensions of kg·m^2·s^2, not torque.I α1/2: taking a square root of alpha lacks physical basis and breaks dimensionality.Common Pitfalls:
Confusing linear and angular forms (F = m * a vs tau = I * alpha).Forgetting that radians are dimensionless, which keeps units consistent.Final Answer:
I α
Discussion & Comments