Final answer:
By differentiating the angular velocity equation with respect to time, we directly obtain the angular acceleration. The equation for rotational dynamics, τ = Iα, relates torque to angular acceleration and moment of inertia, serving as an analog to Newton's second law.
Step-by-step explanation:
To demonstrate that by differentiating the expression for angular displacement with respect to time (t), we can obtain the equation of motion for angular dynamics, we need to start with the kinematic equations for rotational motion. For a body with constant angular acceleration (a), its angular velocity (w) at any time can be expressed as w = w0 + at, where w0 is the initial angular velocity. Differentiating this equation with respect to t gives us the angular acceleration directly, as the derivative of angular velocity with respect to time is the angular acceleration.
The equation for rotational dynamics is given as τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration. This is analogical to Newton's second law for linear motion which states F = ma (F is the force, m is the mass, a is the acceleration). Here, torque is the rotational equivalent of force, angular acceleration is the rotational equivalent of linear acceleration, and the moment of inertia is analogous to mass in the context of rotational dynamics. Therefore, by differentiating the angular displacement over time, we establish the relationship between torque, moment of inertia, and angular acceleration, which is the familiar form of Newton's second law for rotation.