Oscillations 4.1
 A periodic motion is one during which a body continually retraces its path at equal intervals
 It’s motion is continually changing, periodically, such that it reaches it’s maximum and minimum positions
returning to its original position in regular time periods.
 Examples: child’s swing, a person bouncing on a trampoline, bungee jumper, springboard’s vibration after
the diver has left the board, electrons in an antenna
 Frequency
 The frequency of motion, f, is the rate of repetition of the motion -- the number of cycles per unit time.
There is a simple relation between frequency and period:
 Angular frequency
 Angular frequency is the rotational analogy to frequency. Represented as ω(omega) , and is the rate of
change of an angle when something is moving in a circular orbit. This is the usual frequency (measured in
cycles per second), converted to radians per second. That is
 Oscillations
 Displayed below is a position-time graph of a piston moving in and out.
 Phase
 In-phase – time for two particles to reach max and min is equal
 Out-of-phase- time for two particles to reach max and min is NOT equal
 Phase
 Measured in either degree or radians
 In-phase – exactly one time period apart, one full wavelength, 360º or 2π radians
 Out-of-phase – not full period apart, not a full wavelength,
 Roundabout
 Child appears to be moving back and forth or in simple harmonic motion.
 SHM
 Characterized by an acceleration vector always directed toward the equilibrium position and directly
proportional to the (-)displacement.
 ω = 2π / T
 Phase Difference
 The angle by which one oscillation lags behind or leads in front of another oscillation.
 a = -ω2x
 a = aceeleration
 ω = angular velocity
 x = radius of circular motion
 Practice Questions 4.1
 The diagram below shows a mass M suspended from a vertically supported spring. The mass is pulled down
to the position marked A and released such that it oscillates with SHM between the positions A and B. The
equilibrium poition of the mass is at the labeled position E.
 The time period of the oscillation is 0.8s with an max acceleration of 4m/s2.
 Draw a graph that shows how the acceleration of the mass varies with time over one time period.
 Mark on the graph all the points that correspond to the positions A, B, and E on the diagram.
 Practice Question 4.2
 A particle is undergoing simple harmonic motion. When it is passing through its equilibrium position, which
one of the following about it’s acceleration and kinetic energy is correct?
 A. zero accel. and max KE
 B. zero accel. and zero KE
 C. max accel. and zero KE
 D. max accel. and max KE
 Practice Question 4.3
 A mass on the end of a spring undergoes simple harmonic motion about an equilibrium position as shown.
(Have Mr. B draw it.)
 If the upward direction is taken as positive, which graph best represents how the acceleration of the mass
varies with displacement from the equilibrium position? (Have Mr. B draw it.)
 Practice Question 4.4
 When an object undergoes SHM, which of the following is true of the magnitude of the acceleration of the
object?
 A. It is uniform throughout the motion?
 B. It is greatest at the end points of the motion.
 C. It is greatest at the midpoint of the motion.
 D. It is greatest at the midpoints and the endpoints.
 Practice Questions
 At the equilibrium position (x=0) with the mass moving upwards, it has it’s maximum velocity in the
upwards (positive) direction. At the highest point, the velocity is zero, but the mass feels its maximum
downward acceleration. In the lowest point, the velocity is zero again and the mass feels it’s maximum
upward acceleration.
 Phase
 Here is an oscillating ball.
 Its motion can be described as follows:
 Then it moves with v < 0 through the center to the left
 Then it is at v = 0 at the left
 Then it moves with v > 0 through the center to the right
 Then it repeats...
 Simple harmonic motion is defined as the motion that takes place when the acceleration, a, of an object is
always directed towards, and proportional to, its displacement from a fixed point. This acceleration is
caused by a restoring force that must always be pointed towards the mean position and also proportional to
the displacement from the mean position.

F
-X or F=-(constant) x X
 Since F=ma

a
-X or a=-(constant) x X
 The negative sign signifies that the acceleration is always pointing back towards the mean position.
 The constant of proportionality between acceleration and displacement is often identified as the square of a
constant ω which is referred to as the angular frequency.
 a=-ω2x
 Points to note about Simple Harmonic Motion:
 The time period T does not depend on the amplitude A.
 Not all oscillations are simple harmonic motion, but there are many everyday examples of natural simple
harmonic motion.
 Watch the oscillating duck. Let's consider velocity now
 Remember that velocity is a vector, and so has both negative and positive values.
 Watch the oscillating duck. Let's consider acceleration now
 Remember that acceleration is a vector, and so has both negative and positive values.
 Equations
 x = x0 cos(ωt)
(t=0 where x = x0)
 x = x0 sin(ωt)
(t=0 where x = 0)
 v = v0 cos(ωt)
(t=0 where v = max)
 v = v0 sin(ωt)
(t=0 where v = 0)
 v = ω √(x02 – x2) (v = instantaneous velocity)
 Lets draw a graph for each situation
 Practice Problem 4.5
 The graph shows the variation with time t of the displacement, x of a particle undergoing simple harmonic
motion.
 Which graph correctly shows the variations with time, t of the acceleration a of the particle?