15.4 Equations of Simple Harmonic Motion

Question 1

What is the basic defining equation for shm?

Answer 1

a = -ω^2x 
where  
a = acceleration 
ω = angular velocity 
x = displacement

Question 2

What does a graph of acceleration against displacement look like ?

Answer 2

Reflected graph along y-axis of IV graph for Ohms law.

This is because there is a negative in the equation

Question 3

Is the a/x graph infinite?

Answer 3

No it stops at maximum displacement

Question 4

What is the equation for the restoring force?

Answer 4

F = -kx

Question 5

What is the equation for restoring force similar as?

Answer 5

Hooke’s Law

Question 6

What is the equation for the displacement of object moving with shm?

Answer 6

x = Acos(ωt) 
where 
x = displacement 
A = amplitude(max displacement) 
ω = angular velocity/frequency  
t = time

Question 7

What is the equation for ω (the shm constant)

Answer 7

ω = 2πf

Question 8

What kind of a graph is a displacement over time graph?

Answer 8

A cosine graph

Question 9

Hence what is the graph of velocity over time for an object oscillating with shm?

Answer 9

An inverse sine graph

Question 10

What is velocity given by?

Answer 10

v = -Aωsin(ωt) 
where  
v = velocity 
A = amplitude 
ω  = angular frequency/velocity 
t = time

Question 11

How can we get velocity from displacement equation?

Answer 11

By differentiation

Question 12

What is velocity in the displacement time graph?

Answer 12

the gradient

Question 13

What is the equation for acceleration?

Answer 13

a = -Aω^2cos(ωt)

Question 14

What is force proportional to and hence what is acceleration proportional to?

Answer 14

Force is proportional to displacement and since F is proportional to a, a must be proportional to x. The constant is -omega^2

Question 15

How can we find the angular frequency (ω) using the constant k?

Answer 15

if F = ma and F = -kx then ma = -kx
rearranging this gives a = -kx/m
since a = -ω^2
k/m = ω^2 so ω = (square root)k/m

Question 16

If the displacement was started at midpoint (x = 0) then what would we need to do?

Answer 16

NOTE all equations including trigonometric functions apply if the displacement started was at max. If it was at midpoint we swap all sines and cosines.

Question 17

How can we obtain the graphs for displacement, velocity and acceleration?

Answer 17

Have a spring with masses which have a reflective card at the bottom. Motion sensor is placed below the masses and the device is connected to a data logger which is connected to a computer. This can also be done with a rotary motion sensor at the pivot with a data logger.

Question 18

What are the points we absolutely need to remember for calculations?

Answer 18

• amplitude is half the distance between peaks

Question 19

What is the equation for period?

Answer 19

T = 2π / ω

Question 20

What is the equation for period for simple pendulums?

Answer 20

T = 2π (square root)length/g

Question 21

Equation for ω (both equations)

Answer 21

ω = 2π f or ω = 2π /T

Question 22

equation for frequency?

Answer 22

F = ω/2π

Question 23

What kind of a waveform does alternating current have?

Answer 23

A sinusoidal one, this is the same for pd .

Question 24

How is it a sine graph?

Answer 24

As an a.c generator consists of a cylindrical electromagnet driven by a turbine,rotating at high speeds inside a large conducting coils. Every half a cycle its orientation is switched with the respect to the coil and so induced current changes direction.