Simple Harmonic Motion

Question 1

In SHM, what is the equation for force on a mass connected to a spring which is oscillating?

Answer 1

F = -kx, where k is the spring constant. -ve sign to show that the force always acts in the opposite direction to the displacement.

Question 2

How do you derive the equation for ω, the angular frequency?

Answer 2

Set ma = -kx, so d^2 x/dt^2 = -k/m x. Set k/m = ω^2, so ω = sqrt(k/m)

Question 3

What is the equation for the displacement in SHM?

Answer 3

x = Acos(ωt) + Bsin(ωt)

Question 4

What is the equation for angular frequency in terms of tension?

Answer 4

ω = 2pi/T

Question 5

What equation do you use to derive the SHM displacement equation?

Answer 5

x = cos(λt), and then take the double derivative and put into the equation for ω found before.

Question 6

What is an alternative form of the displacement equation in SHM?

Answer 6

x = A’cos(ωt + φ), where A’ is the amplitude and φ is the phase angle.

Question 7

What can help us to find the constants A and B in the SHM displacement equation?

Answer 7

Having initial conditions such as t=0.

Question 8

What is the equation for the potential energy of a spring?

Answer 8

U = 1/2 * k*x^2

Question 9

What is the relationship between displacement, velocity and acceleration?

Answer 9

Displacement = x
Velocity = dx/dt
Acceleration = d^2 x/dt^2

Question 10

What is e^(iϴ) equal to in terms of trigonometric functions?

Answer 10

e^(iϴ) = cos(ϴ) + i*sin(ϴ)

Question 11

What is the complex form of the SHM displacement?

Answer 11

x = Re(A’e^(i(ωt + φ)))

Question 12

How can we use the complex amplitude a to find the real amplitude and the phase angle?

Answer 12

a = A’*e^(iφ), so x = Re(ae^(iωt)). The modulus of a is the amplitude and the argument is the phase angle.

Question 13

What is the equation of damped SHM?

Answer 13

d^2 x/dt^2 + γ dx/dt + ω^2 x = 0

Question 14

How do you derive the equation of damped SHM?

Answer 14

Take ma = -kx-bdx/dt, where b is some constant multiplied by a velocity, which is the damping. This gives m d^2 x/dt^2 + b dx/dt + kx = 0. Divide through by m and set ω^2 = k/m, and define γ = b/m, you get the equation.

Question 15

How do you anticipate a complex solution for the damped SHM equation?

Answer 15

Change x for z, and set z = a*e^(pt). Then take the derivative and the double derivative and substitute into the equation for damped SHM - this is a quadratic equation for p.

Question 16

What are the 3 regimes of damping?

Answer 16

γ < 2ω - light or under-damping
γ > 2ω - heavy or over-damping
γ = 2ω - critical damping

Question 17

How do you write the roots of the complex form for light damping?

Answer 17

p = -γ/2 +/- i*sqrt(ω^2 - (γ/2)^2), and substitute in ω’ as sqrt(ω^2 - (γ/2)^2).

Question 18

How do you get from the roots of light damping to the displacement of light damping?

Answer 18

z = ae^((-(γ/2) +/- iω’)t)

Expand the exponents brackets and write a = |a|*e^(iφ), and take the real pat.

Question 19

What is the equation for displacement during light damping?

Answer 19

x = |a|*e^(-(γt/2)) * cos(ω’t + φ)

Question 20

What does the displacement drop by in time light damping?

Answer 20

Drops by 1/e in a time 2/γ

Question 21

What is the equation for displacement of heavy damping?

Answer 21

x = ae^(p1t) + be^(p2t), where p1 and p2 are the two solutions to the original quadratic equation.

Question 22

Where is critical damping used?

Answer 22

In car shock absorbers and in scientific instruments.

Question 23

What is the equation for the displacement of critical damping?

Answer 23

x = (a+bt)*e^(-(γt/2))

Question 24

What happens when you apply a force F = F0cos(ωt) to the damped mass?

Answer 24

The quadratic equation becomes equal to this force rather than zero. You can divide through by m to get d^2 z/dt^2 + γ dz/dt + ω^2 z = F0/m * e^(iωt)

Question 25

How do you get the equation for amplitude after applying a force?

Answer 25

Set z equal to a*e^(iωt) and differentiate twice. Then substitute into the equation and rearrange for a.

Question 26

How do you convert the complex solution for a after applying a force to a real solution?

Answer 26

Take the magnitude of a by squaring all of the bottom components and then square rooting.

Question 27

What is the equation for the period of an SHM pendulum?

Answer 27

T = 2pi*sqrt(l/g)