Simple Harmonic Motion Flashcards
What is simple harmonic motion?
The motion of an object whose acceleration is proportional to its distance from a fixed point with this acceleration always being directed towards that point.
How is the displacement from the point of equilibrium calculated?
x = Acosωt
How is velocity at a given time calculated?
v = -ωAsinωt
How is the acceleration towards the point of equilibrium calculated?
a = -ω^2Acosωt
How is velocity at a displacement calculated?
v = ±ω√(A^2-x^2)
How is maximum velocity calculated?
Vmax = ±ωA
When does the maximum velocity occur?
At the equilibrium position
When does the minimum velocity occur?
At the maximum displacement
How is maximum acceleration calculated?
a_max = -ω^2A
When does the maximum acceleration occur?
When x = A, at maximum displacement
When does the minimum acceleration occur?
At the equilibrium position
How is time period calculated?
T = 2π/ω
How is frequency calculated?
f = 1/T
What is ω equal to?
2πf
When is the minus sign in an equation ignored?
When asked for the magnitude
What does the gradient of a displacement vs time graph give?
Velocity
What does the gradient of a velocity vs time graph give?
Acceleration
What function does a displacement vs time graph show?
Cosine
What function does a velocity vs time graph show?
-Sine
What function does an acceleration vs time graph show?
-Cosine
What does an acceleration vs displacement graph show?
A straight line with a negative gradient
How is the resulting force of a pendulum calculated?
F = -mgsinθ
What is the acceleration of a pendulum?
a = -gsinθ
How is the time period of a pendulum calculated?
T = 2π√(l/g)
When does sinθ = θ?
When the angle is small and θ is in radians.
How is the maximum velocity of a pendulum calculated?
Vmax = ±A√(g/l)
How is the maximum acceleration of a pendulum calculated?
a_max = -(g/l)A
How is the velocity of a pendulum calculated?
v = ±(√[g/l])(√[A^2-x^2])
What is the restoring force of a spring?
-kx provided Hooke’s Law applies
How is the acceleration of an oscillating spring calculated?
a = -(k/m)x
How is the time period of an oscillating spring calculated?
T = 2π√(m/k)
How can the spring constant of an oscillating spring be found?
k = 4π^2 / gradient
How is the frequency of an oscillating spring calculated?
f = (1/2π)(√[k/m])
How is the maximum velocity of an oscillating spring calculated?
Vmax = ±(√[k/m])A
How is the total energy of an oscillating system calculated?
0.5(mω^2)A^2 or 2mπ^2f^2A^2
How can k be calculated in a system with two springs?
k = F/x
How can the total energy in an oscillating spring system be calculated?
0.5kA^2
What is an undamped system?
An SHM system in which no energy is lost.
How does the amplitude of an undampened system change?
The amplitude is constant.
What is a damped system?
An SHM system in which energy is lost to the surroundings
How does the amplitude of a damped system change?
The amplitude decreases with time.
What is a lightly damped system?
A system in which it takes many oscillation cycles before the amplitude falls to zero.
What is a heavily damped system?
A system in which it the oscillations decay to zero after a few oscillations?
What is an over damped system?
A system which will not oscillate.
What is a critically damped system?
A system which will return to the equilibrium position in less than a quarter of the periodic time.
What are free oscillations?
SHM with a constant amplitude and time period where there is no energy transfer.
What are damped oscillations?
SHM in which the amplitude decreases due to damping forces in which energy is transferred to the surroundings.
What are forced oscillations?
SHM that is driven by an external influence.
What is resonance?
When one oscillating system is coupled to and producing forced oscillations in a second oscillating system.
What is the natural frequency of a system?
The frequency at which a system oscillates when not subjected to a continuous or repeated external force.
When is a responder resonating?
When the frequency of the driver matches the natural frequency of the responder.
What is the phase difference between a responder and driver when there is resonance?
π/2
What is the phase difference between a responder and driver when the driver is below the responder’s natural frequency?
~0
What is the phase difference between a responder and driver when the driver is above the responder’s natural frequency?
~π
How can the effects of resonance be decreased?
Damping
Describe how simple harmonic motion in a spring can be shown experimentally.
- Assemble a clamp and stand on a workbench.
- Attach a spring with an IR position sensor alongside a metre rule.
- Add masses to the spring then pull the assembly down a set amount giving the initial amplitude.
- The masses will now oscillate in SHM as will be shown by the output of the position sensor.
How does damping affect resonance?
- Lightly damped systems will have a sharp resonance peak.
- Heavily damped systems are less sensitive to the driving frequency.
