Simple harmonic motion

Question 1

What is SHM?

Answer 1

The oscillations about a fixed point when an object is displaced and a resulting restoring force acts to oppose the displacement

Question 2

What is the equation for force for a mass on a spring?

Answer 2

F=-kx
Where k is the spring constant and x is the extension

Question 3

What is the general form of SHM?

Answer 3

d^2x/dt^2+ω^2 x=0
Where ω^2=k/m

Question 4

What is the general solution for the SHM equation

Answer 4

x=Acos(ωt)+Bsin(ωt)

Question 5

How does frequency link to ω?

Answer 5

f=ω/2π

Question 6

How does the period link to ω?

Answer 6

T=1/f=2π/ω

Question 7

What is the alternate general solution to the simple SHM?

Answer 7

x=A’cos(ωt+φ)

Question 8

What is the work done in SHM?

Answer 8

W=∫kx’dx’(from 0 to x)=1/2 kx^2

Question 9

With x=0 as the zero potential what is the potential energy in SHM?

Answer 9

U=1/2 kx^2

Question 10

What is the total energy in SHM?

Answer 10

1/2kA^2

Question 11

What is the kinetic energy for SHM?

Answer 11

T=1/2mv^2=1/2m(dx/dt)^2

Question 12

What is the average kinetic energy?

Answer 12

<T>=1/4kA^2
</T>

Question 13

What is the average potential energy?

Answer 13

<u>=1/4kA^2</u>

Question 14

What is Euler’s formula?

Answer 14

Euler’s formula e^(iθ)=cosθ+isinθ

Question 15

What is the complex form of SHM?

Answer 15

z=ae^(iωt) is a solution of d^2z/dt^2+ω^2 z=0
Where a is the complex amplitude A’e^(iθ)

Question 16

What is the equation of damped SHM?

Answer 16

d^2x/dt^2+ω^2 x+γdx/dt ̇=0 where γ=b/m
And d^2z/dt^2+ω^2 z+γdz/dt ̇=0 in complex notation

Question 17

What is the equation for p?

Answer 17

p=-γ/2±√((γ/2)^2-ω^2 )

Question 18

What are the three types of damping and when do they occur?

Answer 18

If γ<2 ω p is complex, oscillation, decay
If γ>2 ω p is real and negative, decay, no oscillation
If γ=2ω p=- γ/2 which is a special case

Question 19

What is the displacement for light damping?

Answer 19

X=Re(z)=|a| e^(-γt/2) cos⁡(ω^’ t+Φ)

Question 20

What is the general solution for heavy damping?

Answer 20

z=ae^(p_1 t)+be^(p_2 t)

Question 21

What is the SHM equation for forced oscillations?

Answer 21

d^2z/dt^2+γdz/dt+ω_0^2 z=F_0/m e^iωt

Question 22

What is the complex a for forced oscillations?

Answer 22

a=(F_0⁄m)/((ω_0^2-ω^2 )+iγω)

Question 23

What is the modulus of the complex amplitude and the phase?

Answer 23

|a|=(F_0⁄m)/√((ω_0^2-ω^2 )+(γω)^2 ) and tanΦ=(-γω)/(ω_0^2-ω^2 )

Question 24

What is the equation of motion for a simple pendulum?

Answer 24

d^2θ/dt^2=-gsinθ/l
Which goes to
d^2θ/dt^2=-gθ/l for small θ

Question 25

What is the angular velocity ω for a simple pendulum?

Answer 25

ω^2=g/l

Question 26

What is the time period for a simple pendulum?

Answer 26

T=2π√(l/g)