Section 3.3: Conservation of Mechanical Energy

The principle of conservation of mechanical energy states that for an isolated system with only conservative forces acting, the total mechanical energy remains constant:

\( E_\text{mechanical} = KE + PE = \text{constant} \)

Where \( KE = \frac{1}{2} m v^2 \) is kinetic energy and \( PE \) is potential energy (gravitational or elastic).

Implications:
  • If only conservative forces act, mechanical energy is conserved: \( KE_i + PE_i = KE_f + PE_f \).
  • If non-conservative forces like friction act, mechanical energy decreases, and work done by these forces converts energy to other forms.

Example 1

A 2 kg ball is dropped from a height of 5 m. Find its speed just before hitting the ground.

Initial mechanical energy: \( E_i = PE = m g h = 2 * 9.8 * 5 = 98 \text{ J} \)

Just before hitting ground: \( KE = E_i \Rightarrow \frac{1}{2} m v^2 = 98 \Rightarrow v = \sqrt{\frac{2*98}{2}} = 9.9 \text{ m/s} \)

Example 2

A 0.5 kg block is compressed against a spring with k = 200 N/m by 0.1 m and then released. Find the speed of the block when the spring returns to equilibrium.

Energy stored in spring: \( PE_{spring} = 0.5 * 200 * 0.1^2 = 1 \text{ J} \)

Block's KE at equilibrium: \( KE = \frac{1}{2} m v^2 = 1 \Rightarrow v = \sqrt{\frac{2*1}{0.5}} = 2 \text{ m/s} \)

Practice Problems

  1. A 3 kg ball falls from 4 m. Find speed just before hitting ground.
  2. A 2 kg block slides down a frictionless incline of height 6 m. Find velocity at bottom.
  3. A 0.5 kg mass compresses a spring k=100 N/m by 0.2 m. Find speed when released.
  4. A pendulum of length 1 m is released from 30° angle. Find speed at bottom assuming small-angle approximation.
  5. A 5 kg object slides down a frictionless ramp from 10 m. Find KE at bottom.
  6. Mass = 1 kg dropped from 10 m onto spring k=500 N/m. How much is spring compressed if it stops the mass?
  7. A roller coaster of 500 kg at 20 m height. Speed at bottom neglecting friction?
  8. A mass of 4 kg slides down a ramp of height 3 m. What fraction of energy is KE at halfway down?
  9. A block compresses spring by 0.15 m storing 2 J. Find spring constant.
  10. Mass 2 kg slides from 5 m height with 10% energy lost to friction. Find speed at bottom.
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