Oscillations

Question 1

Understand and be able to use these terms: displacement, amplitude, period, frequency, angular frequency, phase difference

Answer 1

displacement - from the equilibrium position
amplitude - the maximum displacement
period - the time taken for one full oscillation
frequency - the number of oscillations per unit time
angular frequency - same as angular velocity from topic 12 circular motion
phase difference - how out of sync two oscillations are
I THINK

Question 2

The period of an oscillation can be expressed in terms of both _________ and _______ _________

Answer 2

frequency, angular frequency

Question 3

Simple harmonic motion (SHM) occurs when …

Answer 3

acceleration is proportional to displacement from a fixed point and in the opposite direction

Question 4

What is an oscillation?

Answer 4

a repetitive back-and-forth or up-and-down motion

Question 5

What is natural frequency?

Answer 5

the frequency at which a body vibrates when there is no (resultant external) resistive force acting on it

Question 6

What is the difference between free and forced oscillations

Answer 6

Free oscillations occur at an object’s natural frequency;
Forced oscillations occur when there is a resultant external resistive force acting on a vibrating body
I THINK

Question 7

Give the equations for instantaneous displacement in SHM

Answer 7

Instantaneous displacement (y) = Amplitude (A) x sin{angular frequency (ω) x time (t)}
ω = 2(pi)/T so y = Asin{(2(pi)t/T)
or y = Asin{2(pi)Ft)
*use cos instead of sin if y = A at t = 0 –> ie. timing starts at extreme position

Question 8

Derive two equations for instantaneous velocity in SHM
(these equations ARE provided at the front of exams)

Answer 8

use derivation on the equation y = Asin(ωt)
to get to v = Aωcos(ωt)
And:
v = Aωcos(ωt)
v = ω√{A²cos²(ωt)}
v = ω√{A²[1-sin²(ωt)]}
v = ω√{A²-A²sin²(ωt)}
v = ω√{A²-[Asin(ωt)]²}
v = ω√{A²-y²}

Question 9

Derive two equations for instantaneous acceleration in SHM
(the second equation IS provided at the front of exams)

Answer 9

use derivation of the equation v = Aωcos(ωt)
to get to a = -Aω²sin(ωt) and a = -ω²y

Question 10

Analyse and interpret graphical representations of the variations of displacement, velocity and acceleration for simple harmonic motion

Question 11

At what point in an oscillation is Vmax and how is it calculated
(also do for Vmin)

Answer 11

Vmax occurs at displacement zero, at the lowest point of a swinging pendulum oscillation, and is calculated as +/- Aω
Vmin is, of course, zero - and is at the highest point of a swinging pendulum oscillation, at maximum displacement (amplitude)

Question 12

Describe the interchange between kinetic and potential energy during simple harmonic motion

Answer 12

Energy changes forms continuously between KE and PE and back again, while total energy remains the same - for an undamped oscillation
Graphically (energy on y-axis against displacement on x-axis), it would be shown as a positive quadratic from negative amplitude to positive amplitude, a symmetrical negative quadratic, and a straight horizontal line at the peak of the negative quadratic - showing that total energy remains the same throughout

Question 13

Recall and use an equation for the total energy of a system undergoing simple harmonic motion

Answer 13

E = (1/2)m(Aω)²
using the formula for KE with Vmax

Question 14

Derive an equation for kinetic energy of a system at any position during simple harmonic motion

Answer 14

KE = (1/2)m[ω√{A²-y²}]²
KE = (1/2)mω²(A²-y²)

Question 15

How would you find the potential energy at any position during simple harmonic motion (derive an equation)

Answer 15

PE = TE - KE
PE = (1/2)m(Aω)² - (1/2)mω²(A²-y²)
PE = (1/2)mA²ω - (1/2)mω²A² - (1/2)mω²y
PE = (1/2)mω²y

Question 16

A _________ force on an oscillating system causes _______.

Answer 16

resistive, damping

Question 17

Differentiate between light, critical and heavy damping

Answer 17

Heavy = lots of resistance = slow (don’t think heavy = fast)
Light = less resistance = fast
Critical = just enough resistance to cause the oscillation to stop in the shortest possible time, with no overshooting the equilibrium position

Question 18

What are the four main points of damping

Answer 18

1. amplitude of oscillation keeps decreasing
2. the peaks become flatter
3. the frequency at which resonance occurs shifts to a lower value (moves left on a graph)
4. graphically, a damped oscillation shows an exponential decrease, where the time period and frequency remains the same

Question 19

Resonance involves a maximum _________ of oscillations and that this occurs when an oscillating system is forced to oscillate at ___ _________ _________

Answer 19

amplitude, its natural frequency