This cheat sheet summarizes the period formulas for simple pendulums and mass-spring systems, two of the most common oscillators in high school physics. Students need it because period, frequency, mass, length, gravity, and spring constant are often mixed up in word problems. It helps connect the formulas to the physical factors that make oscillations faster or slower.
A simple pendulum has period when the angle is small, so length and gravitational field strength matter most. A horizontal or vertical mass-spring system has period , so mass and spring stiffness determine the timing. Frequency is related by , and angular frequency is related by .
Key Facts
- The period of a simple pendulum for small angles is , where is length and is gravitational field strength.
- The period of an ideal mass-spring oscillator is , where is mass and is the spring constant.
- Frequency and period are reciprocals, so and .
- Angular frequency is .
- For a simple pendulum, increasing the length increases the period because .
- For a spring oscillator, increasing the spring constant decreases the period because .
- For a spring oscillator, increasing the mass increases the period because .
- The small-angle pendulum formula works best when the initial angle is about or less.
Vocabulary
- Period
- The period is the time required for one complete cycle of oscillation.
- Frequency
- Frequency is the number of cycles per second and is measured in hertz, where .
- Angular frequency
- Angular frequency describes how quickly the oscillator moves through its cycle in radians per second.
- Simple pendulum
- A simple pendulum is an ideal mass on a light string that swings under gravity with period for small angles.
- Spring constant
- The spring constant measures spring stiffness and appears in Hooke's law as .
- Amplitude
- Amplitude is the maximum displacement from equilibrium during an oscillation.
Common Mistakes to Avoid
- Using mass in the simple pendulum period formula, because does not depend on the bob's mass for an ideal pendulum.
- Using amplitude in the ideal spring period formula, because does not depend on amplitude when Hooke's law is valid.
- Forgetting to use SI units, because should be in meters, in kilograms, in newtons per meter, and in seconds.
- Confusing frequency with period, because means a larger period gives a smaller frequency.
- Applying the simple pendulum formula at large angles, because assumes small-angle motion.
Practice Questions
- 1 A pendulum has length on Earth where . Find its period .
- 2 A mass is attached to a spring with . Find the period of the oscillator.
- 3 An oscillator has period . Find its frequency and angular frequency .
- 4 A student doubles the length of a pendulum and doubles the mass of the bob. Explain which change affects the period and why.
Understanding Pendulum & Spring Period Reference
A period is the time for one complete repeating motion. For a pendulum, one cycle starts at one side, passes through the lowest point, reaches the other side, then returns to the starting side. Gravity provides the restoring effect.
When the bob is displaced, gravity has a component that pulls it back toward the lowest position. Near that position, the pull is close to being proportional to the displacement.
This is why the motion is nearly regular for a small release angle. At larger angles, the restoring pull does not match this simple pattern, so each swing takes slightly longer than the reference prediction.
The length of a pendulum means the distance from the pivot to the center of the bob. Students sometimes measure to the bottom or top of the bob, which creates a systematic error. The mass of the bob has almost no effect in the ideal model.
A heavier bob experiences a greater gravitational force, but it also has greater inertia by the same factor. Those effects cancel. A longer pendulum takes more time because its bob travels along a wider arc and its restoring acceleration is less effective for a given angular displacement.
Gravity changes the timing too. A pendulum would swing more slowly on the Moon because the gravitational field is weaker there.
In a spring system, the spring itself supplies the restoring force. Stretching or compressing a spring creates a force toward its equilibrium length. A stiffer spring gives a larger restoring force for the same displacement, so it can reverse the mass more quickly.
The mass keeps moving past equilibrium because of inertia, then the spring pulls it back again. This exchange between kinetic energy and elastic potential energy continues during an ideal oscillation. The amplitude changes the amount of energy stored, but it does not significantly change the timing for an ideal spring.
Real springs can behave differently if stretched too far. They may no longer follow a proportional force rule, or they may become permanently deformed.
Good measurements require timing many cycles instead of one. Start a stopwatch as the object passes a clear reference point, count perhaps ten or twenty complete oscillations, then divide the total time by the number of cycles. This reduces the effect of human reaction time.
Keep the pendulum release angle small and release it without pushing. For a vertical spring, measure displacement from the equilibrium position, not from the spring's unstretched length. Air resistance and friction gradually remove energy, making the amplitude shrink.
Light damping usually changes the period only a little, but strong damping can stop clear oscillations. These ideas appear in playground swings, clock mechanisms, vehicle suspension, seismometers, musical instruments, and sensors.
When solving problems, first identify which object provides the restoring force, then check units carefully. Length is measured in metres, mass in kilograms, spring constant in newtons per metre, and period in seconds.