Section 1.3: Work and Energy (Advanced)

This section covers advanced concepts of work, kinetic energy, potential energy, and the work-energy theorem, including variable forces and systems of particles.

Key Equations & Concepts:
  • Work: \( W = \int \vec{F} \cdot d\vec{s} \) for variable force
  • Kinetic Energy: \( K = \frac{1}{2} m v^2 \)
  • Potential Energy: \( U = m g h \) or spring: \( U = \frac{1}{2} k x^2 \)
  • Work-Energy Theorem: \( W_{net} = \Delta K \)
  • Conservation of Mechanical Energy: \( K_i + U_i = K_f + U_f \) (no non-conservative work)

Example 1

A 5 kg block is pushed along a horizontal surface by a variable force \( F(x) = 2x \) N from x = 0 to x = 3 m. Find the work done by the force.

\( W = \int_0^3 F(x) dx = \int_0^3 2x dx = [x^2]_0^3 = 9 \text{ J} \)

Example 2

A 2 kg mass slides down a frictionless incline of height 5 m. Find its speed at the bottom.

Potential energy lost: \( U = m g h = 2 \cdot 9.8 \cdot 5 = 98 \text{ J} \)

Kinetic energy gained: \( K = \frac{1}{2} m v^2 = 98 \implies v = \sqrt{\frac{2 K}{m}} = \sqrt{98} \approx 9.9 \text{ m/s} \)

Practice Problems

  1. A spring (k = 200 N/m) is compressed 0.1 m. Find stored potential energy.
  2. A 3 kg object moves under force \( F = 5x \) N from x = 0 to x = 4 m. Compute work done.
  3. A 2 kg mass falls from height 10 m. Find speed at bottom ignoring friction.
  4. A block moves under friction μ = 0.1 over 5 m with F = 20 N. Compute net work.
  5. A pendulum of length 2 m is released from 30°. Find speed at lowest point.
  6. A 4 kg block slides down incline h = 3 m, v = 0 initially. Find kinetic energy at bottom.
  7. Variable force \( F(x) = 3x^2 \) N over 0 ≤ x ≤ 2 m. Find work done.
  8. Spring k = 500 N/m, stretched 0.2 m. Compute force at maximum stretch and work done.
  9. A roller coaster drops 20 m. Find final speed using energy conservation.
  10. Mass 5 kg pushed by constant force 10 N over 4 m. Compute work and final speed.
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