Module 5: C17 - Oscillations Flashcards
What happens to an object that is displaced from its equilibrium position
It travels towards the equilibrium position at increasing speed. It then slows down once it has gone past the equilibrium position and eventually reaches maximum displacement (amplitude). It then returns towards its equilibrium position, speeding up, and once more slows down to a stop when it reaches maximum negative displacement. This motion is repeated over and over again.
Displacement
-Symbol
-Unit
-Definition
Symbol: x
Unit: m, Meters
Definition:
The distance from the equilibrium position
Amplitude
-Symbol
-Unit
-Definition
Symbol: A
Unit: m, Meters
Definition:
The maximum displacement from the equilibrium position
Period
-Symbol
-Unit
-Definition
Symbol: T
Unit: s, Seconds
Definition:
The time taken to complete one full oscillation
Frequency
-Symbol
-Unit
-Definition
Symbol: f
Unit: Hz, Frequency
Definition:
The number of complete oscillations per unit time.
What is Simple Harmonic Motion (SHM)
Simple Harmonic Motion (SHM) is a common kind of oscillating motion defined as oscillating motion for which the acceleration of the object is given by:
a = -w^2 x
Equation for acceleration of an object in Simple Harmonic Motion
a = -w^2 x
What are 2 key features of objects moving in Simple Harmonic Motion
- The acceleration a of the object is directly proportional to its displacement x, that is a∝x.
- The minus sign in the equation ‘a = -w^2 x ‘ means that the acceleration of the object acts in the direction opposite to the displacement (it returns the object to the equilibrium position).
Does the period of something in SHM depend on the amplitude (and what happens when amplitude increases)
The period of a simple pendulum moving in SHM does not depend on the amplitude of the swing, so the period does not change. Such an oscillator is referred to as an isochronous oscillator.
What two things must an object be undergoing to be what’s described as Simple Harmonic Motion
- The acceleration of the object is towards a fixed point. In other words the acceleration is in the opposite direction to the displacement.
- The acceleration is proportional to the displacement of the object.
What type of graphs do oscillating objects usually produce
Most examples of oscillation produce graphs in a sinusoidal shape and shows the amplitude A, and period T.
How is the change in velocity shown in a graph (+ when is velocity greatest and lowest on the graph)
The change of velocity over time is given by the gradient of the displacement-time graph
- The velocity is greatest when the gradient of the displacement-time graph is greatest (i.e zero displacement)
- The velocity is zero when the gradient of the displacement-time graph is zero (i.e maximum displacement)
How is the change in acceleration shown in a graph (+ when is velocity greatest and lowest on the graph)
The change of acceleration over time is given by the gradient of the velocity-time graph.
- The acceleration is greatest when the gradient of the velocity-time graph is greatest (i.e zero velocity (greatest displacement in the opposite direction))
- The acceleration is zero when the gradient of the velocity-time graph is zero (i.e Maximum velocity (zero displacement)).
Example Question:
A simple pendulum oscillates in SHM with an amplitude of 32mm. It takes 20s to complete 10 complete oscillations.
Calculate:
a) the Frequency
b) the Initial Acceleration
A = 32x10^-3 = 3.2x10^-2m
t = 20s
n = 10 oscillations
20/10 = 2s per oscillations
a)
f = 1/T => 1/2
f = 0.5Hz
b)
a = -ω^2x
a = -(2π/T)^2 x
a = - (2π/T)^2 x 32x10^-3
a = -0.32ms^-2
What is the displacement x of an object in SHM given by
x = Acosωt (for displacement A at time t=0)
x = Asinωt (for zero displacement at time t=0)
Where A is the maximum displacement
Or
x = Acos(2πft)
x = Asin(2πft)
What is the velocity of an object in SHM given by?
The velocity of an object in SHM is given by the gradient of a displacement time graph. This is the differential dy/dx of x = Asinωt.
The solution is v = ωA cos (ωt)
(Where v is velocity, t is time, ω is angular frequency, A is max. displacement)
What is the maximum velocity of an object in SHM
v = ωA cos (ωt)
Note this will be a maximum when x = 0 and cos(ωt) = 1.
Therefore:
Vmax = ωA = 2πfA
Equation for Displacement, Velocity, and Acceleration of an object in SHM
Displacement = Asin(ωt)
Velocity = ωAcos(ωt)
Acceleration = -ω^2Asin(ωt)
(a = -ω^2x)
How can you calculate the velocity of a simple harmonic oscillator at displacement x? (And also when x=0)
v = ± ω √(A^2 - x^2)
When displacement x = 0
Vmax = ωA