Welcome to Unit 7: Oscillations!
In this unit, we explore things that wiggle, vibrate, and swing back and forth. This is called Periodic Motion. Whether it’s a guitar string, a pendulum in a grandfather clock, or even the atoms inside a crystal, the physics is remarkably similar. Don't worry if this seems a bit math-heavy at first; once you see the patterns, it becomes one of the most predictable and satisfying parts of physics!
1. Defining Simple Harmonic Motion (SHM)
Not every repeating motion is Simple Harmonic Motion (SHM). For a system to be in SHM, it must have a restoring force that is directly proportional to its displacement from equilibrium and directed toward that equilibrium position.
The "Fingerprint" of SHM
The mathematical requirement for SHM is that the acceleration must be proportional to the negative of the displacement:
\( a \propto -x \).
In AP Physics C, we express this as a second-order differential equation:
\( \frac{d^2x}{dt^2} = -\omega^2 x \)
Whenever you see an equation where the second derivative of a variable is equal to negative some-constant times the variable itself, you are looking at SHM!
Key Terms:
• Equilibrium: The "rest" position where the net force is zero.
• Amplitude (A): The maximum distance from equilibrium.
• Period (T): The time it takes for one full cycle (measured in seconds).
• Frequency (f): How many cycles happen per second (measured in Hertz, Hz).
• Angular Frequency (\(\omega\)): The rate of oscillation in radians per second (\(\omega = 2\pi f = \frac{2\pi}{T}\)).
Key Takeaway:
For SHM to occur, there must be a force trying to pull the object back to the center, and that force must get stronger the further away the object moves.
2. The Equations of Motion
If we solve the differential equation mentioned above, we get functions that describe where the object is at any time. We usually use cosine or sine because they naturally repeat themselves.
Position, Velocity, and Acceleration
• Position: \( x(t) = A \cos(\omega t + \phi) \)
• Velocity: \( v(t) = -A\omega \sin(\omega t + \phi) \)
• Acceleration: \( a(t) = -A\omega^2 \cos(\omega t + \phi) \)
Wait, what is \(\phi\)?
The Phase Constant (\(\phi\)) tells us where the object was at time \( t = 0 \). If you start the clock while the object is at its maximum positive position, \( \phi = 0 \).
Memory Aid: The "Speed" Trick
To find the maximum values for velocity and acceleration, just look at the coefficients in front of the trig functions:
• \( v_{max} = A\omega \)
• \( a_{max} = A\omega^2 \)
Quick Review:
- Velocity is zero at the endpoints (where \( x = \pm A \)).
- Velocity is maximum at the equilibrium position (where \( x = 0 \)).
- Acceleration is maximum at the endpoints (where the force is strongest).
3. Mass-Spring Systems
One of the most common examples of SHM is a mass attached to a spring. This follows Hooke’s Law: \( F_s = -kx \).
Period of a Spring
The period depends only on the mass (\( m \)) and the spring constant (\( k \)):
\( T = 2\pi \sqrt{\frac{m}{k}} \)
Note that the amplitude does NOT affect the period. If you pull the spring further, it has a longer distance to travel, but it also moves faster, so the time stays the same!
Did you know?
Even if a spring is hanging vertically, the period stays the same as if it were horizontal. The only difference is that the equilibrium "rest" position is shifted slightly due to gravity.
Common Mistake to Avoid:
Students often flip the fraction inside the square root. Remember: "More Mass = More Time" (since mass is in the numerator, increasing it increases \( T \)).
4. Energy in SHM
In an ideal system (no friction), total mechanical energy is conserved. It constantly sloshes back and forth between Potential Energy (\( U \)) and Kinetic Energy (\( K \)).
• Total Energy: \( E_{total} = \frac{1}{2}kA^2 \)
• Potential Energy: \( U_s = \frac{1}{2}kx^2 \)
• Kinetic Energy: \( K = \frac{1}{2}mv^2 \)
At the endpoints: All energy is Potential (\( U = E_{total} \)).
At the equilibrium: All energy is Kinetic (\( K = E_{total} \)).
Key Takeaway:
Total energy is proportional to the square of the amplitude. If you double the amplitude, you quadruple the energy!
5. Pendulums
There are two types of pendulums you need to know for Physics C: Simple and Physical.
The Simple Pendulum
This is a "point mass" on a string of length \( L \). It only acts like SHM for small angles (usually less than 15 degrees).
\( T = 2\pi \sqrt{\frac{L}{g}} \)
Notice that the mass does not matter for a simple pendulum!
The Physical Pendulum
This is a real-world object (like a swinging baseball bat) rotating around a pivot point.
\( T = 2\pi \sqrt{\frac{I}{mgd}} \)
• \( I \): Rotational inertia of the object about the pivot.
• \( m \): Total mass.
• \( d \): Distance from the pivot to the Center of Mass.
Analogy:
Think of the Simple Pendulum as a "special case" of the Physical Pendulum. If you treat a point mass at distance \( L \) using the physical pendulum formula, it simplifies perfectly to the simple pendulum formula!
Step-by-Step for Physical Pendulum Problems:
1. Identify the pivot point.
2. Find the moment of inertia \( I \) about that pivot (you might need the Parallel Axis Theorem!).
3. Find the distance \( d \) from the pivot to the center of mass.
4. Plug them into the formula.
6. Final Tips for Success
• Check your Calculator: Most SHM problems involve trig functions with \(\omega t\). Ensure your calculator is in Radians mode, not Degrees!
• Look for the Form: If a problem asks you to "prove the motion is simple harmonic," your goal is to show that \( \frac{d^2(variable)}{dt^2} = -(constant)(variable) \).
• Graphs are Key: Be prepared to identify graphs of position vs. time. If it starts at the max, it's a cosine graph. If it starts at zero, it's a sine graph.
You've got this! Oscillations can be tricky because they involve calculus and trigonometry, but the core idea is just a steady, repeating cycle of energy exchange. Keep practicing those differential equations!