Section 5.8: Advanced Problems
Advanced rotational dynamics problems combining torque, angular momentum, and conservation principles. These challenge conceptual understanding and problem-solving skills.
Example 1
A solid sphere rolls down an incline of 30° without slipping. Determine its linear acceleration and rotational speed at the bottom.
Linear acceleration: \( a = \frac{g \sin\theta}{1 + k} = \frac{9.8 \sin30°}{1 + 2/5} \approx 3.53 \text{ m/s²} \)
Rotational speed at bottom: \( \omega = v/R \) with \( v = \sqrt{2 a h} \)
Example 2
A disc with I = 2 kg·m², initially at rest, is subjected to a torque of 5 N·m for 4 s. Calculate final angular velocity and rotational kinetic energy.
Angular acceleration: \( \alpha = \tau/I = 5/2 = 2.5 \text{ rad/s²} \)
Final ω: \( \omega_f = \alpha t = 2.5 * 4 = 10 \text{ rad/s} \)
Rotational KE: \( K = 0.5 I \omega_f^2 = 0.5*2*100 = 100 \text{ J} \)
Practice Problems
- A solid cylinder rolls down a 45° incline. Find linear acceleration and ω at the bottom.
- A thin hoop of mass 5 kg, radius 0.3 m, starts from rest with torque 6 N·m. Find ω after 2 s.
- Two disks, I1 = 2 kg·m² and I2 = 3 kg·m², rotate with ω1 = 10 rad/s and ω2 = 0. Find ω after they are locked together.
- A figure skater spins at ω = 4 rad/s, I = 2 kg·m², arms out to I = 3 kg·m². Find new ω and rotational KE change.
- Wheel I = 1 kg·m², torque 2 N·m applied for 5 s. Calculate θ turned and ω final.
- Flywheel, I = 4 kg·m², spins at ω = 6 rad/s. Friction torque 0.5 N·m applied. Find time to stop.
- Rod, length 2 m, mass 3 kg, pivoted at end. Apply torque 10 N·m for 3 s. Compute ω final.
- Disk, radius 0.5 m, I = 0.25 kg·m², torque 4 N·m, t = 2 s. Compute angular displacement θ.
- A pulley, I = 0.2 kg·m², rope exerts force 5 N at radius 0.1 m for 3 s. Find ω_f.
- Solid sphere rolls up incline with initial v = 5 m/s. Determine max height reached considering rotation.