Simple Harmonic Motion Flashcards
In SHM, what is the equation for force on a mass connected to a spring which is oscillating?
F = -kx, where k is the spring constant. -ve sign to show that the force always acts in the opposite direction to the displacement.
How do you derive the equation for ω, the angular frequency?
Set ma = -kx, so d^2 x/dt^2 = -k/m x. Set k/m = ω^2, so ω = sqrt(k/m)
What is the equation for the displacement in SHM?
x = Acos(ωt) + Bsin(ωt)
What is the equation for angular frequency in terms of tension?
ω = 2pi/T
What equation do you use to derive the SHM displacement equation?
x = cos(λt), and then take the double derivative and put into the equation for ω found before.
What is an alternative form of the displacement equation in SHM?
x = A’cos(ωt + φ), where A’ is the amplitude and φ is the phase angle.
What can help us to find the constants A and B in the SHM displacement equation?
Having initial conditions such as t=0.
What is the equation for the potential energy of a spring?
U = 1/2 * k*x^2
What is the relationship between displacement, velocity and acceleration?
Displacement = x Velocity = dx/dt Acceleration = d^2 x/dt^2
What is e^(iϴ) equal to in terms of trigonometric functions?
e^(iϴ) = cos(ϴ) + i*sin(ϴ)
What is the complex form of the SHM displacement?
x = Re(A’e^(i(ωt + φ)))
How can we use the complex amplitude a to find the real amplitude and the phase angle?
a = A’*e^(iφ), so x = Re(ae^(iωt)). The modulus of a is the amplitude and the argument is the phase angle.
What is the equation of damped SHM?
d^2 x/dt^2 + γ dx/dt + ω^2 x = 0
How do you derive the equation of damped SHM?
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.
How do you anticipate a complex solution for the damped SHM equation?
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.
What are the 3 regimes of damping?
γ < 2ω - light or under-damping
γ > 2ω - heavy or over-damping
γ = 2ω - critical damping
How do you write the roots of the complex form for light damping?
p = -γ/2 +/- i*sqrt(ω^2 - (γ/2)^2), and substitute in ω’ as sqrt(ω^2 - (γ/2)^2).
How do you get from the roots of light damping to the displacement of light damping?
z = ae^((-(γ/2) +/- iω’)t)
Expand the exponents brackets and write a = |a|*e^(iφ), and take the real pat.
What is the equation for displacement during light damping?
x = |a|*e^(-(γt/2)) * cos(ω’t + φ)
What does the displacement drop by in time light damping?
Drops by 1/e in a time 2/γ
What is the equation for displacement of heavy damping?
x = ae^(p1t) + be^(p2t), where p1 and p2 are the two solutions to the original quadratic equation.
Where is critical damping used?
In car shock absorbers and in scientific instruments.
What is the equation for the displacement of critical damping?
x = (a+bt)*e^(-(γt/2))
What happens when you apply a force F = F0cos(ωt) to the damped mass?
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)
How do you get the equation for amplitude after applying a force?
Set z equal to a*e^(iωt) and differentiate twice. Then substitute into the equation and rearrange for a.
How do you convert the complex solution for a after applying a force to a real solution?
Take the magnitude of a by squaring all of the bottom components and then square rooting.
What is the equation for the period of an SHM pendulum?
T = 2pi*sqrt(l/g)
