Section 8.2: Energy in SHM

In Simple Harmonic Motion, the total mechanical energy is conserved and is a combination of kinetic and potential energy.

\[ E_{\text{total}} = K + U = \frac{1}{2} m v^2 + \frac{1}{2} k x^2 \] where:
\( E_{\text{total}} \) = total mechanical energy (J)
\( K \) = kinetic energy (J)
\( U \) = potential energy stored in spring (J)
\( m \) = mass (kg)
\( v \) = speed (m/s)
\( k \) = spring constant (N/m)
\( x \) = displacement (m)

Key points:

  • Maximum potential energy occurs at maximum displacement (\( x = \pm A \)).
  • Maximum kinetic energy occurs at equilibrium position (\( x = 0 \)).
  • Total energy remains constant: \( E_{\text{total}} = \frac{1}{2} k A^2 \).

Example: Energy in a Mass-Spring System

A 0.5 kg mass attached to a spring (\( k = 200 \, \text{N/m} \)) oscillates with amplitude 0.1 m. Find the total energy, kinetic energy at \( x = 0.05 \, \text{m} \), and potential energy at the same position.

Total energy: \( E_{\text{total}} = \frac{1}{2} k A^2 = \frac{1}{2} \cdot 200 \cdot 0.1^2 = 1 \, \text{J} \)
Potential energy at \( x = 0.05 \, \text{m} \): \( U = \frac{1}{2} k x^2 = 0.25 \, \text{J} \)
Kinetic energy: \( K = E_{\text{total}} - U = 1 - 0.25 = 0.75 \, \text{J} \)

Practice Problems

  1. A 1 kg mass attached to a spring of constant 100 N/m oscillates with amplitude 0.2 m. Calculate total energy and energies at \( x = 0.1 \, \text{m} \).
  2. A mass-spring system has total energy 5 J and amplitude 0.25 m. Find the kinetic energy at \( x = 0.15 \, \text{m} \).
  3. A 0.3 kg mass oscillates with maximum speed 2 m/s. Calculate total energy and spring constant if amplitude is 0.1 m.
  4. For a mass-spring system with \( k = 50 \, \text{N/m} \) and \( A = 0.3 \, \text{m} \), find potential and kinetic energy at \( x = 0.2 \, \text{m} \).
  5. A spring with \( k = 200 \, \text{N/m} \) has a 0.5 kg mass attached and total energy 1 J. Determine maximum velocity of the mass.
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