How would a graph show that an object is undergoing SHM?
- The graph models acceleration against displacement
- The graph produces a straight line through the origin, meaning that acceleration is directly proportional to the displacement
- The gradient is negative, meaning that acceleration is directed towards the point of equilibrium
What are the characteristics of SHM?
- Displacement is maximum at extremes of oscillation
- Velocity is greatest at the midpoint
- The oscillating mass is in equilibrium at the midpoint of its oscillation
- When the mass oscillates its inertia carries it through the midpoint
What is the main characteristic of SHM?
- Motion is isochronic I.e. the period is independent of the amplitude
What is the “restoring force”?
The resultant force created from a displacement which acts towards the point of equilibrium. Displacement is proportional to the restoring force,
F = -x, F = -kx
Which formulas are used in terms of sine and cos to calculate displacement?
x = A sin 2pi ft
x = A sin w t
Or
x = A cos 2pi ft
x = A cos w t
A = amplitude
f = frequency
t = Time period
w = angular velocity
Which formulas are used to calculate velocity? How does this relate to the displacement?
v = wA cos w t
When x = A sin w t
v = -wA sin w t
When x = A cos w t
A = amplitude
w = angular velocity
t = time period
x = displacement
How do you calculate the maximum velocity during SHM?
Vmax = -Aw
A = amplitude
w = angular velocity
How do you calculate the maximum acceleration?
a max = -Aw^2
A = amplitude
w = angular velocity
Which formulas are used to calculate acceleration? How does this relate to displacement?
a = -w^2 A sin w t
Or
a = -w^2 A cos w t
a = -w^2 x
A = amplitude
w = angular velocity
t = time period
x = displacement
What are free oscillations?
Undamped oscillations in which if the object is displaced and released, it will oscillate at its own natural frequency with no further input of energy
Give examples of some physical properties which determine natural frequency
Mass, length, spring stiffness
Define damping
Where the amplitude of oscillations is reduced by external forces. This can be used to get rid of unwanted vibrations
What are the three types of damping
Light damping
Heavy damping
Critical damping
Describe light damping
An object will eventually be brought back to rest at equilibrium and will continue to oscillate during this time. This takes the highest amount of time.
Describe heavy damping
An object will be brought back to rest at equilibrium and will not oscillate again. The time taken to bring the oscillator back to equilibrium is much longer than light damping.
Describe critical damping
The minimum amount of damping required to bring an oscillator back to equilibrium without oscillation
What are forced oscillations?
When one oscillating object forces another to oscillate at the frequency of the forced oscillator. This involves a driver and a driven oscillator.
In which condition does resonance occur? What does this result in?
When the driving frequency matches the natural frequency of the driven oscillator, resulting in a maximum transfer of energy from the driver to the driven and a maximum amplitude.
A sphere is suspended across 5 floors on a tower to dampen movement of wind and earthquakes. Why would the natural frequency need to be similar to the frequency of the tower?
To ensure maximum energy is transferred from the building to the sphere.
Why do horizontally displaced objects experience SHM?
The resultant force of the spring is directly proportional to the displacement
How does damping relate to velocity?
Damping is always the opposite of velocity
How does damping relate to displacement?
Damping is maximum when the displacement is 0.
For an object undergoing horizontal SHM using 2 springs, why is the magnitude of the resultant force F = 2kx?
- One force is increased by kx and one is reduced by kx
- To find resultant force, use
F = kx - (-kx) (-ive as is in opposite direction)
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