LECTURE OUTLINE Periodic Motion: Simple Harmonic Motion Motion of a Spring-Mass System The Simple Pendulum Mathematical Description of Simple Harmonic Oscillation L–13 Simple Harmonic Motion (SHM) 1
Each group is required to take distance v/s voltage readings (10 readings ) on your wind generator and plot a graph. This is the last week to work on project.
SHM occurs when the net force is proportional to the displacement from the equilibrium point and is always directed toward the equilibrium point. i.e. it obeys Hooke’s law where F is the restoring force, x is the displacement and k is the spring constant (or force constant) 3
Examples of SHM - Vibrational motion of atoms and molecules in solids (amplitude depends on temperature). - Simple and Compound pendula; Not every periodic motion is SHM. However most periodic motions approximate to SHM for small displacements. Also assume no friction etc. - Mass on the end of a horizontal or vertical spring; 4
A mass on a frictionless surface is attached to the end of a horizontal spring as shown. The mass is pulled to one side and then released (or compressed to the other side and then released). Proof 1: Show that the subsequent motion is SHM. The equation of motion is: F = -kx = ma x (where k is the force constant of the spring). And so it is SHM. 2. Horizontal motion of a spring-mass system 5
Prove that a mass attached to the end of an ideal vertical spring undergoes SHM when displaced from the equilibrium position. 6
3. The Simple Pendulum A simple pendulum also exhibits periodic motion The motion occurs in the vertical plane and is driven by gravitational force Proof 3 : Prove that, for small angles, the motion of a pendulum approximates SHM. 7
Basic equation of motion for SHM It is often convenient to write this as: where This is a differential equation: ( ω is the angular frequency) 4. Mathematical Description of Simple Harmonic Oscillation 8
Proof 4: Prove that general solution for the equation has a form: Constants A and B depend on the initial conditions i.e. value of x and v when t = 0. 9 The another possible form is φ – initial phase
10 10 Initial conditions at t = 0 are: x (0) = 0 v (0) = v max The graphs of velocity and acceleration are shifted one-quarter cycle to the right compared to the graph above it, where x (0) = 0 Graph of x(t) = A sin( ω t)
11 11 Initial conditions at t =0 : x (0) = A v (0) = 0 Graph of x(t)=A cos( ω t) It is similar to expression x(t)=A sin ( ω t+ π /2) with initial phase of π /2
Physical meaning of this solution At t = 0 then x = 0, so the motion “starts” when the mass is passing through the equilibrium position x varies sinusoidally and the motion repeats with period and frequency ω is the angular frequency A is called the amplitude and is the maximum displacement. The maximum speed is v max = A ω The maximum acceleration is a max = A ω 2 12
For a spring Then And For a simple pendulum Then And 13
A mass of 100 g is on a horizontal spring of constant k = 10.0 Nm -1. It is initially displaced by 0.100 m, and then released so that it oscillates. Assume friction is negligible. A) Find the amplitude, the period and the frequency of the SHM; B) Write down expressions for displacement, velocity and acceleration of SHM as functions of time. 14
An object in SHM has an amplitude A = 0.200 cm and a frequency f = 20.0 Hz. Write down expression for displacement as a function of time taking equilibrium point as initial position (x=0 at t=0); B) Find the velocity and acceleration of an object at x = 0; x = A and x = - A. 15
Ｖ elocity v and acceleration a in terms of displacement acceleration velocity Find v 2 in terms of displacement, not time 16
There is a striking similarity between SHM and uniform circular motion 17
Animations for Simple and physical pendulum See animations at: http://wps.aw.com/aw_young_physics_11/13/3510/898588.cw/nav_and_content/index.html Animation 9.10 : Shows independence of period on mass, and dependence on length Animation 9.12 : A comparison between a simple and a uniform compound pendulum of same mass m. Do they have the same period? If no, which one do you expect to have the shortest period? 18
LECTURE CHECK LIST Understand what is meant by a differential equation (DE); Be able to solve the DE for SHM by using a general solution; Be able to solve problems using the equations of Simple Harmonic Motion. 19
ANSWERS Example 1 0.100 m; 0.628 s; 1.59 Hz; x=(0.100 m) cos( 10.0t) v=-(1.00 m/s) sin( 10.0t ) a=-(10.0 m/s 2 ) cos(10.0t) Example 2 x=(0.200cm)sin(12.6t); 0.251 m/s, 0 and 0; 0, -31.8 m/s² and 31.8 m/s² 20
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