〰️Energy in SHM
Energy Exchange in SHM
In simple harmonic motion, energy continuously converts between kinetic and potential energy, but the total mechanical energy remains constant.
For a spring-mass system: - At maximum displacement (x = ±A): all energy is spring PE, v = 0 - At equilibrium (x = 0): all energy is KE, v is maximum - In between: energy is shared between KE and spring PE
Total energy: E = ½kA² = ½mv_max²
This gives us the maximum speed: v_max = A√(k/m) = Aω
Energy at Any Position
At any position x in SHM:
KE = ½m v² = ½k(A² − x²) PE = ½kx² Total = ½kA²
At position x: v = ω√(A² − x²)
Notice: at x = A/2, PE = ½k(A²/4) = E/4. KE = 3E/4. The speed at x = A/2 is not half the maximum speed — it's faster!
✏️ Worked Example
Problem: A mass on a spring (k = 100 N/m) oscillates with amplitude 0.2 m. Find (a) total energy, (b) maximum speed if m = 0.5 kg, (c) speed when x = 0.1 m.
📐 Key Equations
Energy in SHM
E_total = (1)/(2)kA² = (1)/(2)mv_max²v = ω√(A² - x²)v_max = Aω = A√((k)/(m))⚠️ Common Mistakes
Misconception: Assuming the speed at x = A/2 is half the maximum speed.
✓ Correct thinking: At x = A/2, KE = ¾E_total, so the speed is √(¾) ≈ 0.866 of v_max — not ½. Speed is not proportional to distance from the endpoint.
Why: v = ω√(A²−x²). At x = A/2: v = ω√(A²−A²/4) = ω√(3A²/4) = (√3/2)Aω ≈ 0.866 v_max.
Misconception: Thinking total energy depends on both mass and spring constant separately.
✓ Correct thinking: Total energy E = ½kA² depends only on k and amplitude. For finding v_max, you also need mass: v_max = A√(k/m). But E itself doesn't require m.
Why: Energy is stored in the spring as PE = ½kx². At maximum stretch (x = A), all energy is spring PE = ½kA².
Misconception: Using conservation of energy incorrectly by forgetting that PE_spring = ½kx², not mgh.
✓ Correct thinking: For spring-mass systems, use PE = ½kx². Gravitational PE (mgh) is only relevant for pendulums or vertically oscillating springs where height changes matter.
Why: Spring potential energy stores energy in the compression/extension of the spring. Gravitational PE stores energy based on vertical height. Don't mix them unless both are present.
📝 Practice Problems
Try these problems. Check your answer when ready.
A spring (k = 80 N/m) oscillates with amplitude 0.25 m. What is the total mechanical energy?
A 0.4 kg mass on a spring has total energy 5 J. What is its maximum speed?
A mass-spring system (k = 50 N/m, m = 0.2 kg) oscillates with A = 0.1 m. Find the speed when x = 0.06 m.
v = ω√(A² - x²)A spring-mass system has E_total = 8 J and k = 200 N/m. What is the amplitude of oscillation?
At what position x (as a fraction of A) does a spring-mass oscillator have equal kinetic and potential energy?
KE = PE ⇒ (1)/(2)k(A² - x²) = (1)/(2)kx²A 0.5 kg mass on a spring (k = 180 N/m) is released from x = 0.2 m with an initial velocity of 1 m/s in the direction toward equilibrium. Find the amplitude of the resulting oscillation.
E_total = (1)/(2)mv² + (1)/(2)kx²Finished reading through this lesson?