SIMPLE HARMOIC MOTION CCHS Physics

Facts of SHM SHM occurs when an object is vibrating at a single frequency and period PERIODOIC: when a vibration or oscillation repeats itself, back and forth, over the same path We’ll deal mainly with simple harmonic oscillations where the position of the object can be described by a sinusoidal (sin, cos) function

Examples of SHM Block attached to spring Motion of a pendulum Vibrations of a stringed musical instrument Clocks Oscillations of houses, bridges, …

Meaning of Simple Harmonic Motion SIMPLE  single frequency HARMONIC  sinusoidal A system in which the restoring force is proportional to the negative displacement (Hooke’s Law) will move in simple harmonic motion.

Spring and SHM

Example of Complex Harmonic Motion Approximately simple-harmonic in this region

Hooke’s Law The magnitude of the restoring force is proportional to the displacement Where: F = force (N) k = spring constant (N/m) x = displacement Acceleration Force and Acceleration Proportional to x Directed toward the equilibrium position

Hooke’s Law cont. Accurate as long as there is not too much displacement There is a negative sign because the force always acts opposite to the displacement Note: F varies with position

Five Fun Definitions DISPLACEMENT: the distance x from the equilibrium point AMPLITUDE: the maximum displacement CYCLE: one complete to-and-from motion PERIOD (T): time to complete one cycle FREQUENCY (f): number of cycles per second Frequency is measured in s-1 = Hertz (Hz)

Period and Frequency Period and Frequency are inversely related:

Force vs. Distance Graphs Slope = spring constant Area = work (or energy)

Energy of a Spring Remember that potential energy is stored energy due to position The calculation of potential energy is equivalent to calculating work Note: usually going to neglect mass of spring

Example 1 Given the following graph: What is the spring constant? If a 2 kg mass attached to this spring is displaced 3 m, what is the acceleration of the mass? If the spring is stretched to 5 m, how much energy is stored in the spring?

Conservation of Energy Mechanical Energy - the sum of all kinetic and potential energies Now we add in elastic potential energy We still have that energy must be conserved, thus:

Example 2 First, find the spring constant: A spring stretches by 5 cm when a 100 g mass is hung from it. A 250 g mass attached to this spring and pulled back 0.70 m and released from rest. What is the speed of the block as it passes through the equilibrium point? First, find the spring constant: Now, use conservation of energy

Example 3 Note: x2 = h1 call this distance d Conservation of Energy A 0.2-kg ball is attached to a vertical massless spring as shown. The spring constant of the spring is 28 N/m. The ball, supported initially so that the spring is neither stretched nor compressed, is released from rest. In the absence of air resistance, how far does the ball fall before being brought to a momentary stop by the spring? Conservation of Energy Note: x2 = h1 call this distance d

Velocity and Position Initially the mass is at its maximum extension A Entirely elastic potential energy The initial elastic potential energy converted to a combination of kinetic and eleastic potential energy Solving for v

Simple Harmonic Motion vs. Uniform Circular Motion A ball is glued to the top of a turntable Focus on the shadow of the ball If the turntable rotates with constant angular velocity the shadow of the ball moves in simple harmonic motion

SHM vs. UCM cont. If we can prove that the velocity of the shadow varies with position like the function on the previous slides, we know the shadow moves in SHM. Looking at the top triangle (made with velocity vectors) Now looking at the larger triangle Equating these two equations Thus we have proved that the velocity of the shadow in the x direction is related to displacement in exactly the same manner as the velocity of an object undergoing SHM.

Frequency and Period The velocity of the ball in the previous slide is: or Where A is the amplitude T is the period Focus on 1/4 of the trip, Imagine the shadow is a block on a spring, in the 1/4 of a cycle the block moves from a point of all elastic potential energy to a point of all kinetic energy Substituting for A/v0

Period of a Pendulum Restoring force = mgsin Not SHM because F  sin, not  However, for small : sin ≈ (measured in radians) Thus, F = -mgsin ≈ mg Also, s = L Essentially SHM (Hooke’s Law) Plugging into period of a spring formula with k = (mg)/L

Displacement Equation Looking at reference circle again Ball starts on the x axis at x = +A and moves through the angle  in a time t Ball rotates at constant angular speed  (because UCM)  = t Displacement x of the shadow is just the projection of the radius A onto the x-axis

Properties of SHM Position of particle at time t: A…amplitude w…Angular frequency f…phase constant, phase angle T…period (wt + F)…phase

Period and Frequency (Revisited) PERIOD: time required to complete one cycle The value of T depends on the angular speed (frequency)  Greater , less T For 1 cycle,  = 2, and t = T: Using the fact that frequency and period are inverses:

Velocity and Acceleration

More Properties of SHM Phase of velocity differs by – p/2 or 90° from phase of displacement. Phase of acceleration differs by – p or 180° from phase of displacement.

Even More Properties of SHM Acceleration of particle is proportional to the displacement, but is in the opposite direction (a = - 2·x). Displacement, velocity and acceleration vary sinusoidally. The frequency and period of the motion are independent of the amplitude.

Summary of SHM t x v a KE Us A -w2A ½kA2 T/4 -wA T/2 -A 3T/4 T

Example A block whose mass m is 680 g is fastened to a spring whose spring constant k is 65 N/m. The block is pulled a distance of x = 11 cm from its equilibrium position at x = 0 on a frictionless surface and released from rest at t = 0. What force does the spring exert on the block just before the block is released? What are the period, frequency and angular frequency?

Example cont. What is the amplitude of the oscillation? The amplitude is 11 cm (that is where it is released from) What is the maximum speed of the oscillating block? max speed occurs whenever x = 0 What is the magnitude of the maximum acceleration of the block? max acceleration occurs at the end points

Example cont. What is the phase constant  of the motion? At t = 0, the displacement is at its maximum value A, and the velocity of the block is zero. If we put these initial conditions into the displacement and velocity equations we find: The smallest angle that solves these two equations is  = 0. (Any multiple of 2 will also work)