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Simple Harmonic Motion (SHM) cheat sheet - grade 10-12

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Physics Grade 10-12

Simple Harmonic Motion (SHM) Cheat Sheet

A printable reference covering SHM displacement, velocity, acceleration, period, frequency, springs, pendulums, and energy for grades 10-12.

TopicsPhysicsMotionForcesNewton's LawsAccelerationEnergyTime
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Simple harmonic motion describes repeated back-and-forth motion caused by a restoring force that is proportional to displacement. This cheat sheet helps students recognize SHM in springs, pendulums, vibrations, and waves. It is useful because many physics problems use the same core relationships for position, velocity, acceleration, period, and energy.

Grades 10-12 students need these formulas to connect graphs, equations, and physical motion.

Key Facts

  • Simple harmonic motion occurs when the restoring force is proportional to displacement and opposite in direction, so F=−kxF = -kx for a spring.
  • Displacement in SHM can be modeled by x(t)=Acos⁡(ωt+ϕ)x(t) = A\cos(\omega t + \phi) or x(t)=Asin⁡(ωt+ϕ)x(t) = A\sin(\omega t + \phi).
  • Angular frequency is related to frequency and period by ω=2πf=2πT\omega = 2\pi f = \frac{2\pi}{T}.
  • Velocity in SHM has maximum magnitude vmax⁡=Aωv_{\max} = A\omega and can be written as v(t)=−Aωsin⁡(ωt+ϕ)v(t) = -A\omega\sin(\omega t + \phi) for cosine displacement.
  • Acceleration in SHM is proportional to displacement and opposite in direction, so a=−ω2xa = -\omega^2 x.
  • The period of a mass-spring oscillator is T=2πmkT = 2\pi\sqrt{\frac{m}{k}}.
  • For small angles, the period of a simple pendulum is T=2πLgT = 2\pi\sqrt{\frac{L}{g}}.
  • Total mechanical energy in an ideal spring oscillator is constant and equals E=12kA2E = \frac{1}{2}kA^2.

Vocabulary

Amplitude
Amplitude is the maximum displacement from equilibrium, represented by AA.
Equilibrium Position
The equilibrium position is the point where the net restoring force is zero.
Period
Period is the time for one complete cycle of motion, represented by TT.
Frequency
Frequency is the number of cycles per second and is given by f=1Tf = \frac{1}{T}.
Angular Frequency
Angular frequency measures how quickly the oscillator moves through its cycle and is given by ω=2πf\omega = 2\pi f.
Restoring Force
A restoring force is a force that acts toward equilibrium and often has the form F=−kxF = -kx.

Common Mistakes to Avoid

  • Using F=kxF = kx instead of F=−kxF = -kx, because the negative sign shows that the force points opposite the displacement.
  • Confusing frequency and angular frequency, because ff is measured in hertz while ω\omega is measured in radians per second and equals 2πf2\pi f.
  • Assuming acceleration is greatest at equilibrium, because acceleration is actually a=−ω2xa = -\omega^2 x and is zero when x=0x = 0.
  • Using the pendulum formula for large swings, because T=2πLgT = 2\pi\sqrt{\frac{L}{g}} only works well for small angles.
  • Thinking the period of a spring depends on amplitude, because an ideal mass-spring oscillator has T=2πmkT = 2\pi\sqrt{\frac{m}{k}} and amplitude is not in the formula.

Practice Questions

  1. 1 A 0.50 kg0.50\,\text{kg} mass is attached to a spring with k=200 N/mk = 200\,\text{N/m}. Find the period TT of the motion.
  2. 2 An oscillator has amplitude A=0.12 mA = 0.12\,\text{m} and angular frequency ω=8.0 rad/s\omega = 8.0\,\text{rad/s}. Find the maximum speed vmax⁡v_{\max}.
  3. 3 A pendulum has length L=1.6 mL = 1.6\,\text{m} on Earth where g=9.8 m/s2g = 9.8\,\text{m/s}^2. Find its period using the small-angle approximation.
  4. 4 Explain why the acceleration of an object in SHM always points toward equilibrium, even when the object is moving away from equilibrium.

Understanding Simple Harmonic Motion (SHM)

A useful way to understand SHM is to follow one complete cycle. At the farthest point from equilibrium, the object stops for an instant before reversing direction. Its displacement has the greatest magnitude there.

The restoring force and acceleration have their greatest magnitudes too, both aimed toward the center. As the object moves toward equilibrium, it speeds up. At equilibrium, displacement is zero, so the restoring force and acceleration are zero.

The speed is greatest at that moment because the object has built up motion during its trip inward. This sequence repeats on the other side in reverse.

Graphs show the same story from different viewpoints. A displacement versus time graph is a smooth sine-like curve. A velocity versus time graph has the same period, but its peaks occur at different times.

When displacement is at a positive maximum, velocity is zero. When displacement crosses zero while moving in the positive direction, velocity is at its positive maximum. Acceleration is opposite to displacement at every instant.

Students often lose marks by treating these graphs as if their high points happen together. Sketching the motion at the center and at both turning points helps connect each graph to a real moving object.

Energy explains why the speed changes. For a horizontal spring with negligible friction, energy shifts continually between elastic potential energy and kinetic energy. At a turning point, all the energy is stored in the stretched or compressed spring and the mass has no kinetic energy.

At equilibrium, the spring has its least elastic potential energy and the mass has its greatest kinetic energy. A larger amplitude means more total energy.

It does not mean a longer period for an ideal mass and spring. Changing the mass or spring stiffness changes the period, while changing amplitude does not, provided the spring remains within its normal elastic range.

Real oscillators are not perfectly ideal. Air resistance, friction, and internal heating remove mechanical energy. The amplitude then becomes smaller over time.

This is called damping. A playground swing, guitar string, car suspension, ruler clamped to a desk, and building during an earthquake can all oscillate. A repeated external push can make an oscillator respond strongly when the push rate is close to its natural frequency.

This is resonance. It can be useful in clocks and musical instruments, but it can damage structures if vibrations grow too large. For pendulums, the simple period model works best at small angles.

Large swings do not follow the model exactly. Students should state assumptions clearly, check units, and decide whether a problem describes an ideal oscillator or one with energy losses.