Kinetics of Rigid Bodies: Work and Energy

Why it matters

The study of work and energy in the kinetics of rigid bodies is crucial for designing and analyzing mechanical systems such as engines, machinery, and structures. It helps engineers predict how these systems will respond to various forces and movements, ensuring safety and efficiency.

Key ideas

• Work-Energy Principle: This principle states that the work done by all external forces acting on a rigid body is equal to the change in its kinetic energy.
• Kinetic Energy of Rigid Bodies: The kinetic energy of a rigid body in motion is the sum of its translational and rotational kinetic energies.
• Conservation of Energy: In the absence of non-conservative forces (like friction), the total mechanical energy (sum of kinetic and potential energy) of a system remains constant.
• Power: The rate at which work is done or energy is transferred in a system.

Formulas

• T = 1/2·m·v² + 1/2·I·ω²
  • T: Total kinetic energy (Joules)
  • m: Mass of the body (kg)
  • v: Center-of-mass velocity (m/s)
  • I: Mass moment of inertia about the center of mass for planar motion (kg·m²)
  • ω: Angular velocity (rad/s)
• W = ΔT
  • W: Work done by external forces (Joules)
  • ΔT: Change in kinetic energy (Joules)
• P_avg = W/Δt; instantaneous P = dW/dt
  • P: Power (Watts)
  • t: Time (seconds)

Worked example

Given: A solid cylinder of mass 10 kg and radius 0.5 m starts from rest and rolls without slipping, with negligible rolling resistance, down a stationary incline of height 5 m. Calculate the velocity of the cylinder at the bottom of the incline.

1. Calculate the moment of inertia (I) for the cylinder:

   • Formula: I = 1/2·m·r²
   • I = 1/2·10 kg·(0.5 m)² = 1.25 kg·m²

2. Apply conservation of energy:

   • Potential energy at the top = Kinetic energy at the bottom
   • m·g·h = 1/2·m·v² + 1/2·I·ω²
   • 10 kg·9.81 m/s²·5 m = 1/2·10 kg·v² + 1/2·1.25 kg·m²·(v/r)²

3. Solve for v:

   • 490.5 J = 5v² + 0.625(v/0.5)²
   • 490.5 J = 7.5v²
   • v² = 65.4 m²/s²
   • v = √65.4 ≈ 8.087 m/s

Final Answer: 8.09 m/s

Common mistakes

• Forgetting to include both translational and rotational kinetic energy in calculations.
• Misapplying the conservation of energy principle by not accounting for all forms of energy.
• Incorrectly calculating the moment of inertia for different shapes.

For GATE CE

Questions often involve calculating the velocity or energy of a rigid body using the work-energy principle. Practice problems involving different shapes and motions, ensuring a strong grasp of moment of inertia calculations.

Quick check

1. What is the work-energy principle?
2. How do you calculate the kinetic energy of a rotating body?
3. What happens to the total mechanical energy in the absence of non-conservative forces?

Answers: 1. The work done by external forces equals the change in kinetic energy. 2. Sum of translational and rotational kinetic energies. 3. It remains constant.