Section 5.8: Review Examples
This section consolidates key problems on rotational motion, torque, equilibrium, moment of inertia, angular momentum, and rotational energy. Solutions include both calculations and conceptual explanations.
Example 1: Rotational Kinetic Energy of a Cylinder
A solid cylinder of mass 5 kg and radius 0.3 m rolls without slipping at a linear speed of 4 m/s. Find its rotational kinetic energy and total kinetic energy.
Moment of inertia: \( I = \frac{1}{2} m r^2 = 0.5*5*0.3^2 = 0.225 \, \text{kg·m²} \)
Rotational KE: \( KE_{rot} = 0.5*0.225*(4/0.3)^2 \approx 20 \, \text{J} \)
Translational KE: \( KE_{trans} = 0.5*5*4^2 = 40 \, \text{J} \)
Total KE: \( 40 + 20 = 60 \, \text{J} \)
Example 2: Torque and Equilibrium
A 2 m long uniform beam weighing 200 N is supported at one end. A 100 N weight is hung 1.2 m from the support. Find the torque about the support and check if the beam remains in equilibrium.
Torque by beam weight (center of mass at 1 m): \( \tau_{beam} = 200*1 = 200 \, \text{N·m} \)
Torque by weight: \( \tau_{weight} = 100*1.2 = 120 \, \text{N·m} \)
Net torque about support = 320 N·m.
If the support provides an equal counter-torque, equilibrium is maintained.
Practice Problems
- A solid sphere of mass 3 kg and radius 0.2 m rotates at 10 rad/s. Find its rotational kinetic energy.
- A wheel of mass 6 kg and radius 0.4 m rolls without slipping at 3 m/s. Determine translational, rotational, and total kinetic energy.
- A uniform rod of length 2 m and mass 4 kg is pivoted at one end. Calculate the torque due to its weight about the pivot when horizontal.
- A solid cylinder rolls down an incline of height 5 m. Determine its speed at the bottom, assuming no slipping.
- A rotating disc has an angular momentum of 12 kg·m²/s and a moment of inertia of 3 kg·m². Find its angular velocity.