Simple Harmonic Motion Flashcards Preview

Classical Mechanics and Special Relativity > Simple Harmonic Motion > Flashcards

Flashcards in Simple Harmonic Motion Deck (27)
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1
Q

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

A

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

2
Q

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

A

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

3
Q

What is the equation for the displacement in SHM?

A

x = Acos(ωt) + Bsin(ωt)

4
Q

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

A

ω = 2pi/T

5
Q

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

A

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

6
Q

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

A

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

7
Q

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

A

Having initial conditions such as t=0.

8
Q

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

A

U = 1/2 * k*x^2

9
Q

What is the relationship between displacement, velocity and acceleration?

A
Displacement = x
Velocity = dx/dt
Acceleration = d^2 x/dt^2
10
Q

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

A

e^(iϴ) = cos(ϴ) + i*sin(ϴ)

11
Q

What is the complex form of the SHM displacement?

A

x = Re(A’e^(i(ωt + φ)))

12
Q

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

A

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

13
Q

What is the equation of damped SHM?

A

d^2 x/dt^2 + γ dx/dt + ω^2 x = 0

14
Q

How do you derive the equation of damped SHM?

A

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.

15
Q

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

A

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.

16
Q

What are the 3 regimes of damping?

A

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

17
Q

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

A

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

18
Q

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

A

z = ae^((-(γ/2) +/- iω’)t)

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

19
Q

What is the equation for displacement during light damping?

A

x = |a|*e^(-(γt/2)) * cos(ω’t + φ)

20
Q

What does the displacement drop by in time light damping?

A

Drops by 1/e in a time 2/γ

21
Q

What is the equation for displacement of heavy damping?

A

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

22
Q

Where is critical damping used?

A

In car shock absorbers and in scientific instruments.

23
Q

What is the equation for the displacement of critical damping?

A

x = (a+bt)*e^(-(γt/2))

24
Q

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

A

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)

25
Q

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

A

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

26
Q

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

A

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

27
Q

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

A

T = 2pi*sqrt(l/g)