Unit 7 Beta

Oscillations: the one-page sheet

7.1 Defining Simple Harmonic Motion

Simple harmonic motion is periodic motion driven by a restoring force proportional to the displacement from equilibrium and opposite to it (F = −kx); the force and acceleration are zero at equilibrium and largest at the amplitude, and a pendulum at small angles is a close match.

Periodic motion repeats in equal time intervals. Simple harmonic motion (SHM) is the special case caused by a restoring force proportional to the displacement: F = −kx.
The equilibrium position is where the net force is zero. The object moves fastest there, so zero force does not mean zero velocity.
The amplitude A is the largest displacement from equilibrium: half the distance from one turning point to the other.
A simple pendulum at small angles is close to SHM: along its arc the restoring force is about mg·θ, proportional to the angle (in radians).

a restoring force pulls it back toward equilibrium the acceleration is largest at the ends and zero at equilibrium it overshoots, because zero force does not stop it the object slows, stops at the same distance A, and comes back: simple harmonic motion

periodic motion: Motion that repeats itself in equal time intervals, such as a swinging pendulum or a block bouncing on a spring.
restoring force: A force that points back toward an object's equilibrium position whenever the object is displaced from it.
equilibrium position: The position where the net force on an oscillating object is zero; the center of its oscillation.
simple harmonic motion: Periodic motion caused by a restoring force proportional to the displacement from equilibrium and opposite to it, F = −kx.
amplitude: The largest displacement from the equilibrium position during an oscillation, A; half the distance between the turning points.
simple pendulum: A small, heavy bob on a light string swinging under gravity; for small angles its motion is very close to simple harmonic motion.

7.2 Frequency and Period of SHM

The period of a block on a spring is T = 2π√(m/k) and that of a small-angle pendulum is T = 2π√(ℓ/g); neither depends on the amplitude, the pendulum's period does not depend on the bob's mass, and f = 1/T.

The period T is the time for one full cycle; the frequency is f = 1/T, in hertz (cycles per second). Time many cycles and divide to measure T well.
A block on a spring: T = 2π√(m/k). More mass, longer period; stiffer spring, shorter period. No amplitude and no g in it.
A simple pendulum at small angles: T = 2π√(ℓ/g). Longer string, longer period; stronger gravitational field, shorter period. No mass in it.
Functional dependence: four times the mass (or length) doubles the period, because T goes as the square root.

accelerates less for the same restoring force, so it takes longer to turn around: T grows as √m gives a larger restoring force at each position, so the period shrinks as 1/√k means more distance to cover but proportionally more force and speed, so the period does not change has more weight pulling it back and more inertia in exactly the same ratio, so its period does not change either

period of oscillation: The time for one complete cycle of an oscillation, T, in seconds; the frequency is f = 1/T, in hertz.
spring-mass period: The period of a block of mass m on a spring of constant k: T = 2π√(m/k). It does not depend on the amplitude or on g.
pendulum period: The period of a simple pendulum of length ℓ at small angles: T = 2π√(ℓ/g). It does not depend on the bob's mass.

7.3 Representing and Analyzing SHM

SHM is described by x = A cos(2πft) or A sin(2πft); on the graphs, speed is greatest and acceleration zero at equilibrium, speed zero and acceleration greatest at the turning points, and the period is independent of the amplitude.

Position in SHM is x = A cos(2πft) if the object starts at +A, or x = A sin(2πft) if it starts at equilibrium moving in +x. Use radians.
On an x-t graph: the height of a peak is A; the time from peak to peak is T; the slope is the velocity.
Speed is greatest at x = 0 and zero at x = ±A. Acceleration is zero at x = 0 and largest at x = ±A, always pointing toward equilibrium.
The period does not depend on the amplitude: a bigger swing has more distance to cover and covers it faster.

its acceleration is always opposite to its displacement, so the a-t graph is the x-t graph flipped its velocity is zero there: the x-t graph is flat at its peaks and troughs its speed is greatest there: the x-t graph is steepest where it crosses zero the cycle takes the same time: the period does not depend on the amplitude

sinusoidal motion: Motion whose position-time graph is a sine or cosine curve, as in SHM: x = A cos(2πft) or x = A sin(2πft).
period independent of amplitude: In simple harmonic motion the period does not depend on the amplitude: a larger swing is covered at a proportionally larger speed.

7.4 Energy of Simple Harmonic Oscillators

With no friction the mechanical energy of an oscillator is constant: for a block on a spring E = ½kA², all potential at the turning points and all kinetic at equilibrium; doubling the amplitude quadruples the energy.

For a block and spring with no friction, E = K + U_s is constant. Name the system (block + spring) and the condition (no friction) before you use it.
The total energy is set by the amplitude: E = ½kA². At the turning points K = 0; at equilibrium U_s = 0 and K = E.
U_s = ½kx² goes as x², so at x = A/2 the spring holds only a quarter of the energy. Energy graphs against position are parabolas, and the total is a flat line.
For a pendulum, the energy trades between kinetic and gravitational potential energy: mgΔy = ½mv² at the bottom.

all the system's energy is spring potential energy, ½kA² spring energy turns into kinetic energy all the energy is kinetic and the block moves at top speed the total stays ½kA², so the block reaches the same distance A on the other side

energy of a simple harmonic oscillator: The total mechanical energy of an oscillating system, constant with no friction; for a block on a spring it equals ½kA², shared between kinetic and spring potential energy.