SOLUTION MANUAL 1 Solutions to Chapter 1 Student Exercises
1.1 Think back to how you derived the equations of motion for 1D motion with constant velocity and 1D motion with constant acceleration. Now derive the equations of motion for rotational motion with constant angular velocity and rotational motion with constant angular acceleration, shown in Table 1.1. Also, construct graphs for the position, velocity, and acceleration (both angular and linear) as functions of time for all combinations of positive and negative velocity, and positive and negative acceleration. Compare the graphs for the linear and angular quantities as functions of time. Why should you not be surprised that the graphs look the same? Let’s start with the equations for rotational motion with constant angular ⃗ . Because the angular velocity is constant, then the average angular velocity ω ⃗ avg is the same as the instantaneous angular velocity ω ⃗ . That is: velocity ω ∆⃗θ . (1.1) ∆t Realizing that when a quantity is constant its average and instantaneous values are equal is the foundation of such derivations where we do not use calculus. Now, we can easily see that ⃗ ∆t, ∆⃗θ = ω (1.2) ⃗ =ω ⃗ avg = ω
which is the equation of motion with constant angular velocity. Of course, we could derive the same equation using integration. By definition, we know that d⃗θ ⃗ = ⃗ dt. ω ⇒ d⃗θ = ω (1.3) dt 1