AP Physics 1: Simple Harmonic Motion (SHM) - The Night Before 🚀

Hey there, future physics master! Let's get you prepped for SHM. This unit is all about periodic motion, think pendulums and springs. It's a smaller part of the exam (~2-4%), but mastering it will boost your confidence and show off your understanding of forces and energy!

Big Ideas at Play

• Big Idea #3: Force Interactions - Forces describe how objects interact. SHM is a great example of forces at work, like the restoring force of a spring.
• Big Idea #5: Conservation - Energy transforms but is always conserved. In SHM, we see kinetic and potential energy constantly trading places.

🔗 Key Concepts

• Period (T) - Time for one full cycle.
• Amplitude - Max displacement from equilibrium.
• Frequency (f) - Cycles per second.
• Equilibrium Point - The resting position.
• Kinetic Energy (K) - Energy of motion.
• Potential Energy (Ug ,Usp) - Stored energy (gravitational or spring).

🧮 Key Equations

6.1 ⏱️ Period of Simple Harmonic Oscillators

What is a Simple Harmonic Oscillator?

A simple harmonic oscillator (SHO) is anything that moves back and forth with a consistent rhythm. Think of a pendulum swinging or a mass bouncing on a spring. The key is that they follow Hooke's Law: the force pulling them back to the center is proportional to how far they've moved from that center. It's like the system is always trying to get back to its happy place.

Period of an SHO

The period (T) is the time it takes for one full back-and-forth motion. It's like the length of one complete dance move. For a mass-spring system, the period is given by:

$$T = 2\pi\sqrt{\frac{m}{k}}$$

Where:

• T is the period (in seconds)
• m is the mass (in kg)
• k is the spring constant (in N/m)

6.2 ⚡ Energy of a Simple Harmonic Oscillator

Energy Transformation

In SHM, energy is constantly shifting between kinetic energy (K) and potential energy (U). It's like a seesaw, where one goes up, the other goes down. The total energy of the system remains constant (assuming no friction).

• Kinetic Energy (K): Energy of motion. It's max when the object is at its equilibrium position (moving fastest) and zero at the extreme ends of its motion (when it stops momentarily).

$$K = \frac{1}{2}mv^2$$

• Potential Energy (U): Stored energy. For a spring, it's max when the spring is most stretched or compressed (object is farthest from equilibrium) and zero at the equilibrium point.

$$U = \frac{1}{2}kx^2$$

Where:

• k is the spring constant
• x is the displacement from equilibrium

Total Energy

The total energy of the system is the sum of kinetic and potential energy:

$$E_{total} = K + U$$

Visualizing Energy

Imagine a mass on a spring. When it's at its max displacement, it's not moving (K=0), but the spring is stretched (U is max). As it moves towards equilibrium, K increases and U decreases. At equilibrium, K is max and U is zero. This dance continues as the mass oscillates.

🎯 Final Exam Focus

• High-Value Topics: Energy transformations in SHM, period calculations for mass-spring systems, and understanding the relationship between displacement, velocity, and acceleration.
• Common Question Types: Multiple-choice questions on energy conservation, FRQs involving period calculations and energy graphs, and questions that combine SHM with other concepts like forces and energy.
• Time Management: Quickly identify SHM problems. Use equations efficiently. Don't spend too long on one question.
• Common Pitfalls: Confusing period and frequency, not squaring displacement in potential energy calculations, and forgetting the conservation of energy principle.

💪 Last-Minute Tips

• Review Key Equations: Make sure you know the period and energy formulas by heart.
• Visualize: Imagine the motion of a mass on a spring or a pendulum. Think about how energy is changing.
• Practice: Work through a few practice problems to build confidence.
• Stay Calm: You've got this! Trust your preparation and stay focused.

📝 Practice Questions

Practice Question

Multiple Choice Questions

1. A mass-spring system oscillates with a period T. If the mass is doubled, what is the new period? (A) T/2 (B) T/√2 (C) T√2 (D) 2T

2. At what point in the oscillation of a mass-spring system is the kinetic energy maximum? (A) At the maximum displacement (B) At the equilibrium position (C) When the velocity is zero (D) When the acceleration is maximum

3. A spring with a spring constant k is stretched by a distance x. If the displacement is doubled to 2x, what happens to the potential energy stored in the spring? (A) It remains the same (B) It is doubled (C) It is quadrupled (D) It is halved

Free Response Question

A 0.5 kg mass is attached to a spring with a spring constant of 20 N/m. The mass is pulled 0.1 m from its equilibrium position and released. Assume no friction.

(a) Calculate the period of the oscillation. (b) Calculate the total energy of the system. (c) Calculate the maximum velocity of the mass. (d) Sketch a graph of potential energy vs. time for one period of the oscillation.

Scoring Breakdown

(a) Period Calculation (2 points)

• 1 point for using the correct formula: $T = 2\pi\sqrt{\frac{m}{k}}$
• 1 point for correct substitution and answer: $T = 2\pi\sqrt{\frac{0.5}{20}} = 0.99 s$

(b) Total Energy Calculation (2 points)

• 1 point for using the correct formula: $E = \frac{1}{2}kx^2$
• 1 point for correct substitution and answer: $E = \frac{1}{2}(20)(0.1)^2 = 0.1 J$

(c) Maximum Velocity Calculation (3 points)

• 1 point for recognizing that total energy equals max kinetic energy: $E = \frac{1}{2}mv_{max}^2$
• 1 point for correct substitution: 0.1 = \frac{1}{2}(0.5)v_{max}^2
• 1 point for correct answer: $v_{max} = 0.63 m/s$

(d) Potential Energy vs. Time Graph (3 points)

• 1 point for a sinusoidal shape
• 1 point for correct period (0.99s)
• 1 point for max potential energy at t=0 and t=T/2 (0.1 J)

Remember, you've got this! Let's ace that exam! 🌟