7.4 Energy of Simple Harmonic Oscillators

In simple harmonic motion, the total mechanical energy stays constant while energy shifts back and forth between kinetic and potential forms. Kinetic energy peaks at equilibrium, where speed is highest, and potential energy peaks at the turning points, where the object stops.

Why This Matters for the AP Physics 1 Exam

Energy in oscillating systems pulls together work, energy, and conservation ideas from earlier units and applies them to springs and pendulums. On the exam, you may need to track energy as an object oscillates, connect graphs of energy versus position or time, and explain why total energy stays constant. This topic also gives strong practice for the kind of free-response question that asks you to build and compare multiple representations, such as free-body diagrams and energy bar charts for a block on a spring at maximum displacement and at equilibrium. Expect to move between equations, words, and graphs while keeping units and conservation logic consistent.

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Key Takeaways

• Total mechanical energy in SHM is $E_{total} = U + K$ and stays constant for an ideal (frictionless) system.
• Kinetic energy is greatest at the equilibrium position, where displacement is zero and speed is at its maximum.
• Potential energy is greatest at the turning points (maximum displacement), where the object momentarily stops and kinetic energy is zero.
• For a spring-object system, $E_{total} = \frac{1}{2}kA^2$, so energy scales with the square of the amplitude.
• Doubling the amplitude quadruples the total energy because energy depends on $A^2$.
• Real systems lose energy to friction, which damps the oscillation over time.

Mechanical Energy in SHM

Total Energy Components

In a system undergoing simple harmonic motion (SHM), the total energy is the sum of the system's kinetic energy (K) and potential energy (U).

$E_{total} = U + K$

• Kinetic energy comes from the object's motion and depends on its mass and speed.
• Potential energy depends on position and the restoring force acting on the object.
• For a spring, potential energy is stored as elastic potential energy in the stretched or compressed spring.
• For a pendulum, potential energy is stored as gravitational potential energy.

Conservation of Total Energy

Energy conservation means the total energy of an isolated system does not change; it only converts between forms.

$E_{total} = U + K = \text{constant}$

• During oscillation, energy continuously converts between kinetic and potential forms.
• At any instant, the sum of kinetic and potential energy equals the total energy.
• In ideal SHM, total mechanical energy stays constant as energy shifts between K and U.
• In real systems, friction gradually reduces total energy and damps the oscillation.

Maximum Kinetic Energy

A system in SHM reaches maximum kinetic energy when it passes through the equilibrium position.

$K_{max} = \frac{1}{2}mv_{max}^2$

• At equilibrium, displacement is zero and speed is at its maximum.
• All the energy is kinetic at this point.
• The net force on the object is zero at equilibrium.
• This is the moment when the object switches the direction of its acceleration.

Maximum Potential Energy

Potential energy in SHM is greatest at the extreme positions of the oscillation.

At the turning points, potential energy is at its maximum and kinetic energy is zero, so $E_{total} = U_{max}$. For a spring-object system specifically, $E_{total} = U_{max} = \frac{1}{2}kA^2$.

• At maximum displacement (the amplitude), velocity is zero.
• All the energy is potential at these turning points.
• The restoring force reaches its largest magnitude here.
• After reaching maximum displacement, the object reverses direction.

Minimum Kinetic Energy

The kinetic energy of a system in SHM reaches its minimum value of zero at the points of maximum displacement.

• At the turning points, the object momentarily stops before changing direction.
• The velocity at these points is zero, so kinetic energy is zero.
• All the energy has converted to potential form at these points.
• The minimum kinetic energy can be written as $K_{min} = 0$

Effects of Changing Amplitude

For a spring-object system, changing the amplitude changes the maximum potential energy and therefore the total energy. Using $E_{total} = \frac{1}{2}kA^2$, increasing the amplitude increases total energy, and doubling the amplitude quadruples the total energy.

$E_{total} = \frac{1}{2}kA^2$

• Increasing the amplitude increases the maximum potential energy.
• Since total energy equals maximum potential energy, changing the amplitude changes the total energy.
• Doubling the amplitude quadruples the total energy (energy is proportional to amplitude squared).

How to Use This on the AP Physics 1 Exam

Problem Solving

When you work an energy problem in SHM, anchor your reasoning to two special positions:

• At equilibrium ($x = 0$): all energy is kinetic, so $E_{total} = K_{max} = \frac{1}{2}mv_{max}^2$.
• At a turning point ($x = \pm A$): all energy is potential, so $E_{total} = U_{max} = \frac{1}{2}kA^2$.

For any in-between position, find the potential energy with $U = \frac{1}{2}kx^2$, then use $K = E_{total} - U$ to solve for speed. Keep amplitudes and positions in meters so your energy comes out in joules.

Free Response

Some free-response questions ask you to build and compare different representations of the same scenario. For a block oscillating on a spring, you might sketch free-body diagrams at maximum displacement and at equilibrium, then draw energy bar charts for those same points, and finally explain how the two representations agree. To connect them, point out that the largest restoring force lines up with maximum potential energy at the turning point, while zero net force lines up with maximum kinetic energy at equilibrium.

Common Trap

Watch out for amplitude scaling. Because $E_{total} = \frac{1}{2}kA^2$, energy depends on $A^2$, not $A$. If a question doubles the amplitude, the total energy goes up by a factor of four, not two.

Practice Problem 1: Spring Energy Conservation

Solution:

(a) Total energy of the system: $E_{total} = \frac{1}{2}kA^2 = \frac{1}{2} \times 20 \text{ N/m} \times (0.10 \text{ m})^2 = 0.10 \text{ J}$

(b) Maximum speed occurs at equilibrium when all energy is kinetic: $K_{max} = E_{total} = \frac{1}{2}mv_{max}^2$

Solving for $v_{max}$: $v_{max} = \sqrt{\frac{2E_{total}}{m}} = \sqrt{\frac{2 \times 0.10 \text{ J}}{0.5 \text{ kg}}} = 0.63 \text{ m/s}$

(c) Speed at 5 cm from equilibrium: At this position, find how the energy is divided: $U = \frac{1}{2}kx^2 = \frac{1}{2} \times 20 \text{ N/m} \times (0.05 \text{ m})^2 = 0.025 \text{ J}$

Since $E_{total} = K + U$: $K = E_{total} - U = 0.10 \text{ J} - 0.025 \text{ J} = 0.075 \text{ J}$

Speed at this position: $v = \sqrt{\frac{2K}{m}} = \sqrt{\frac{2 \times 0.075 \text{ J}}{0.5 \text{ kg}}} = 0.55 \text{ m/s}$

Practice Problem 2: Amplitude and Energy

Solution:

(a) Total energy with 8 cm amplitude: $E_{total} = \frac{1}{2}kA^2 = \frac{1}{2} \times 15 \text{ N/m} \times (0.08 \text{ m})^2 = 0.048 \text{ J}$

(b) Total energy with 16 cm amplitude: $E_{total} = \frac{1}{2}kA^2 = \frac{1}{2} \times 15 \text{ N/m} \times (0.16 \text{ m})^2 = 0.192 \text{ J}$

(c) Relationship between amplitude and energy: The new total energy (0.192 J) is exactly 4 times the original energy (0.048 J). This shows that energy in a spring-object system is proportional to the square of the amplitude. When amplitude doubles, energy quadruples.

Common Misconceptions

• Total energy is not zero at the turning points. The object stops there, so kinetic energy is zero, but all the energy is now potential, and the total stays the same.
• Kinetic and potential energy are largest at opposite positions. Kinetic energy peaks at equilibrium, while potential energy peaks at the turning points; they are never both at maximum at the same time.
• Changing the amplitude does change the total energy of a spring-object system, because $E_{total} = \frac{1}{2}kA^2$. Amplitude does not change the period, but it does change the energy.
• Energy scales with $A^2$, not $A$. Tripling the amplitude multiplies the total energy by nine, not three.
• The maximum speed in SHM occurs at equilibrium, not at the turning points. At the turning points the speed is zero.

Related AP Physics 1 Guides

• 7.1 Defining Simple Harmonic Motion (SHM)
• 7.3 Representing and Analyzing SHM
• 7.2 Frequency and Period of SHM