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AP Physics 1: Springs & Pendulums

After studying how things move (Kinematics) and why they move (Dynamics), we now look at things that move back and forth. Simple Harmonic Motion (SHM) is the physics of oscillations—specifically, motion driven by a restoring force. In the AP Physics 1 CED this appears in Unit 7: Oscillations, worth about 5–8% of your exam score.

1. What is Simple Harmonic Motion?

Not all vibration is “Simple Harmonic.” For an object to undergo SHM, it must satisfy one specific condition: The restoring force is proportional to the displacement.

$F_{restoring} = -kx$

Hooke’s Law: The farther you pull, the harder it pulls back.

Period vs. Frequency

Before looking at specific objects, you must know the relationship between Period ( $T$ ) and Frequency ( $f$ ).

Period ( $T$ ): The time for one complete cycle (seconds).
Frequency ( $f$ ): The number of cycles per second (Hertz).

$T = \frac{1}{f}$

2. The “Big 2”: Springs vs. Pendulums

The College Board requires you to know exactly which factors change the period for two specific systems.

Split-panel diagram comparing SHM periods. Left: Spring system (depends on mass). Right: Pendulum system (depends on length, mass crossed out).

What affects the Period? For a spring, inertia (mass) increases the period. For a pendulum, only the length and gravity matter—mass is irrelevant.

Key Takeaways:

Springs ( $T_s$ ): Period depends on Mass ( $m$ ) and Spring Constant ( $k$ ).
Think: A heavier mass is harder to accelerate, so it moves slower (longer period). A stiffer spring (high k) pulls back faster (shorter period).
Pendulums ( $T_p$ ): Period depends on Length ( $L$ ) and Gravity ( $g$ ).
Think: A longer string has a wider arc to travel (longer period).

$T_{\text{spring}} = 2\pi \sqrt{\frac{m}{k}}, \quad T_{\text{pendulum}} = 2\pi \sqrt{\frac{L}{g}}$

AP Physics 1: Only these two period formulas are required for SHM on the exam.

⚠️ The “Mass” Trap: If you double the mass of a bob on a pendulum, the period does NOT change. This is the #1 trick question in this unit. Mass cancels out because gravity pulls harder on more massive objects exactly as much as their inertia resists moving.

3. Quick AP Practice

📚 AP Practice Problems

1. A block on a spring oscillates with period $T$ . If the mass is quadrupled ( $4m$ ), what happens to the new period?

Answer

Since $T = 2\pi\sqrt{m/k}$ , quadrupling the mass results in $\sqrt{4} = 2$ . The period doubles ( $2T$ ).

2. You take a pendulum clock to the Moon, where gravity is $1/6$ of Earth’s. Will the clock run fast or slow?

Answer

Lower $g$ means a larger denominator in $T = 2\pi\sqrt{L/g}$ , making the Period $T$ larger (longer). A longer period means the clock takes longer to tick, so it runs slower.

3. A spring–mass oscillator is moved from Earth to a planet where gravity is twice as large. How does the period change?

Answer

For an ideal horizontal spring, $T = 2\pi\sqrt{m/k}$ does not depend on $g$ . The period stays the same.

🔗 Related AP Physics Resources

Part 1 Complete!

You now know how to calculate the period. But what about Velocity, Acceleration, and Energy? That is where the SHM graphs and energy bar charts from the second child page come in.

Next Section: SHM Graphs & Energy →