AP Physics C Mechanics Unit 6 Notes: Learning Simple Harmonic Motion

Simple Harmonic Motion (SHM)

Back-and-forth motion in which acceleration is proportional to displacement from equilibrium and directed toward equilibrium: a = -\omega^2 x.

Equilibrium Position

The position where the net force (or net torque) is zero; SHM displacement x is measured from this point.

Displacement (x) from Equilibrium

The signed position measured relative to equilibrium; in SHM it determines the restoring force/acceleration (e.g., a \propto -x).

Restoring Force

A force that acts to return a system to equilibrium; for SHM it must be (approximately) proportional to displacement and opposite in direction.

Angular Frequency (\omega)

A constant (rad/s) that sets the time scale of SHM; relates to acceleration by a = -\omega^2 x and to period by \omega = \frac{2\pi}{T}.

Hooke’s Law

Spring force is proportional to displacement: F_s = -kx (valid when the spring is not stretched/compressed too far).

Spring Constant (k)

A measure of spring stiffness (N/m) appearing in Hooke’s law; larger k means a “stiffer” spring.

Mass–Spring Oscillator (Horizontal)

A mass m attached to a spring on a frictionless surface that undergoes SHM with \omega = \sqrt{\frac{k}{m}}.

Vertical Spring Equilibrium Extension (x_{eq})

The stretch where the hanging mass is at rest: kx_{eq} = mg; oscillations occur about this shifted equilibrium.

Simple Pendulum

A point mass (bob) on a massless string of length L; for small angles it approximates SHM with \omega = \sqrt{\frac{g}{L}}.

Tangential Component of Gravity (Pendulum)

The restoring force along the arc: F_t = -mg \sin( heta), pointing toward heta = 0.

Torque ( au) About a Pivot

Rotational analog of force that causes angular acceleration; for a pendulum au = -mgL \sin( heta).

Moment of Inertia (I) for a Point Mass Pendulum

For a bob of mass m at distance L from the pivot: I = mL^2.

Angular Acceleration (\alpha)

The second time derivative of angle: $\alpha = \frac{d^2\theta}{dt^2}$; related to torque by $\Sigma \tau = I\alpha$.

Small-Angle Approximation

For sufficiently small heta (in radians), \sin( heta) \approx heta, allowing the pendulum equation to become SHM: heta'' + \frac{g}{L} heta = 0.

Period (T)

Time for one full oscillation (seconds); T = \frac{1}{f} and T = \frac{2\pi}{\omega}.

Frequency (f)

Number of cycles per second (Hz); $f = \frac{1}{T}$ and $f = \frac{\omega}{2\pi}$.

Hertz (Hz)

Unit of frequency meaning s^{-1} (cycles per second).

Phase Constant (\phi)

A constant that shifts the SHM graph in time; set by initial conditions in x(t) = A \cos(\omega t + \phi).

Amplitude (A)

Maximum displacement from equilibrium; sets maximum speed/energy but (for ideal SHM) does not change the period.

SHM Differential Equation

The defining equation of motion: x'' + \omega^2 x = 0 (or heta'' + \omega^2 heta = 0 for small-angle pendulum).

Sinusoidal Solution to SHM

A solution of the form x(t) = A \cos(\omega t + \phi) (equivalently sine), which satisfies x'' = -\omega^2 x.

Maximum Speed (v_{max}) in SHM

Occurs at equilibrium; v_{max} = A\omega (also from energy: \frac{1}{2}kA^2 = \frac{1}{2}mv_{max}^2 for a spring).

Maximum Acceleration (a_{max}) in SHM

Occurs at maximum displacement; a_{max} = A\omega^2 (since a = -\omega^2 x).

Mechanical Energy Conservation in Ideal SHM

With no nonconservative forces, total energy stays constant and shifts between kinetic and potential (spring: $E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$).