simple harmonic motion Flashcards
What are the two conditions required for an object to be in simple harmonic motion?
- The acceleration is proportional to the displacement
- The acceleration is in the opposite direction to the displacement
What are examples of objects that undergo SHM AND how are they periodic?
- The pendulum of a clock
- A mass on a spring
- Guitar strings
- The electrons in alternating current flowing through a wire
These are always periodic, meaning they are repeated in regular intervals according to their frequency or time period
What is a restoring force and its significance in SHM?
- An object in SHM will also have a restoring force to return it to its equilibrium position
- This restoring force will be directly proportional, but in the opposite direction, to the displacement of the object from the equilibrium position
- the restoring force and acceleration act in the same direction
Why is a person jumping on a trampoline not an example of SHM?
- The restoring force on the person is not proportional to their distance from the equilibrium position
- When the person is not in contact with the trampoline, the restoring force is equal to their weight, which is constant
- This does not change, even if they jump higher
What do the components of SHM acceleration equation mean?
What does the SHM acceleration equation demonstrate?
- The acceleration reaches its maximum value when the displacement is at a maximum ie. x = A (amplitude)
- The minus sign shows that when the object is displaced to the right, the direction of the acceleration is to the left and vice versa (a and x are always in opposite directions to each other)
Relationship between SHM acceleration and displacement
directly proportional
What are the features of a displacement-acceleration graphfor an object in SHM
What are the components of the SHM displacement equation
When is displacement at max, according to the SHM displacement equation
- The displacement will be at its maximum when cos(⍵t) equals 1 or −1, when x = A
What is the equation used when an object is oscillating from its equilibrium position (x = 0 at t = 0)
(The displacement will be at its maximum when sin(⍵t) equals 1 or −1, when x = A.
This is because the sine graph starts at 0, whereas the cosine graph starts at a maximum)
when is the greatest speed of an oscillator
when it is in the equilibrium position
What do the components of the SHM speed equation mean
What does the SHM speed equation show
That when an oscillator has a greater amplitude A, it has to travel a greater distance in the same time and hence has greater speed v
Are the displacement, velocity and acceleration graphs for an object in SHM in phase_
No- the displacement, velocity and acceleration graphs in SHM are all 90° out of phase with each other
What are the key features of a displacement-time graph for an object in SHM
- The amplitude of oscillations A can be found from the maximum value of x
- The time period of oscillations T can be found from reading the time taken for one full cycle
- The graph might not always start at 0
- If the oscillations starts at the positive or negative amplitude, the displacement will be at its maximum
What are the key features of a velocity-time graph for an object in SHM
- It is 90 degrees out of phase with the displacement-time graph
- Velocity is equal to the rate of change of displacement
So, the velocity of an oscillator at any time can be determined from the gradient of the displacement-time graph
What are the key features of an acceleration-time graph for an object in SHM
- The acceleration graph is a reflection of the displacement graph on the x axis
- This means when a mass has positive displacement (to the right) the acceleration is in the opposite direction (to the left) and vice versa
- It is 90° out of phase with the velocity-time graph
- Acceleration is equal to the rate of change of velocity
So, the acceleration of an oscillator at any time can be determined from the gradient of the velocity-time graph:
Equation to find v from x and t for SHM
Equation of find a from v and t for SHM
What do the components of the maximum speed equation for SHM mean
When is the maximum speed for a mass on a spring
at the equilibrium position. Its speed is 0 at its positive and negative amplitude
What do the components of the maximum acceleration equation for SHM mean
Why does an object in SHM still have acceleration at maximum amplitude
- Although at the amplitude, the speed is zero, the oscillator has changed direction
- This means that it has a non–zero velocity, and since acceleration is the rate of change of velocity, the oscillator has an acceleration at the amplitude too
At what point does an object in SHM have maximum acceleration
at its positive and negative amplitude. Its acceleration is 0 at the equilibrium position
What do the components of the time period for a mass-spring system equation mean
For what kind of mass-spring systems does the time period equation apply to
What does the mass-spring system time period equation show
-The equation shows that the time period and frequency, of a mass-spring system, does not depend on the force of gravity (the oscillations would have the same time period on Earth and the Moon)
- The higher the spring constant k, the stiffer the spring and the shorter the time period
What do the components of the pendulum time period equation mean
Does the period of a pendulum depend on gravity
yes, meaning its period would be different on the Earth and the Moon
what is an example of SHM which isn’t a pendulum or mass-spring system
liquid in a U-tube
A bungee jumper
An acrobat on a trapeze
A swing
A ball in a concave dish
Oscillating platforms
What could the ‘potential energy’ in SHM refer to
The potential energy could be in the form of:
- Gravitational potential energy (for a pendulum)
- Elastic potential energy (for a horizontal mass on a spring)
- Or both (for a vertical mass on a spring)
when is kinetic energy at a maximum in SHM
when the displacement x = 0 (equilibrium position)
when is potential energy at a maximum in SHM
when the displacement (both positive and negative) is at a maximum x = A (amplitude)
Does the total energy in a SHM system change?
No - it is constant and is equal to the sum of the kinetic and potential energies
What are the key feature of an energy-time graph for an object in SHM
- Both the kinetic and potential energies are represented by periodic functions (sine or cosine) which are varying in opposite directions to one another
- When the potential energy is 0, the kinetic energy is at its maximum point and vice versa
- The total energy is represented by a horizontal straight line directly above the curves at the maximum value of both the kinetic or potential energy
- Energy is always positive so there are no negative values on the y axis
What are the key feature of an energy-displacement graph for an object in SHM
- Displacement is a vector, so, the graph has both positive and negative x values
- The potential energy is always at a maximum at the amplitude positions x = A, and 0 at the equilibrium position x = 0
- This is represented by a ‘U’ shaped curve
- The kinetic energy is the opposite: it is 0 at the amplitude positions x = A, and maximum at the equilibrium position x = 0
- This is represented by an ‘n’ shaped curve
- The total energy is represented by a horizontal straight line above the curves
damping def
The reduction in energy and amplitude of oscillations due to resistive forces on the oscillating system
When does damping stop having an effect on an oscillator
when the oscillator comes to rest at the equilibrium position
what happens to frequency of SHM objects when amplitude decreased
frequency of damped oscillations does not change as the amplitude decreases
For example, a child on a swing can oscillate back and forth once every second, but this time remains the same regardless of the amplitude
What are the types of damping
Light damping
Critical damping
Heavy damping
What is light damping
When oscillations are lightly damped, the amplitude does not decrease linearly
It decays exponentially with time
When a lightly damped oscillator is displaced from the equilibrium, it will oscillate with gradually decreasing amplitude
For example, a swinging pendulum decreasing in amplitude until it comes to a stop
Graph for a lightly damped oscillator
Key features of a displacement-time graph for a lightly damped oscillator
- There are many oscillations represented by a sine or cosine curve with gradually decreasing amplitude over time
- This is shown by the height of the curve decreasing in both the positive and negative displacement values
- The amplitude decreases exponentially
- The frequency of the oscillations remain constant, this means the time period of oscillations must stay the same and each peak and trough is equally spaced
Describe a critically damped oscillator
- When a critically damped oscillator is displaced from the equilibrium, it will return to rest at its equilibrium position in the shortest possible time without oscillating
- For example, car suspension systems prevent the car from oscillating after travelling over a bump in the road
Draw a displacement-time graph for a critically damped oscillator
What are the key features of the displacement-time graph for a critically damped oscillator
- This system does not oscillate, meaning the displacement falls to 0 straight away
- The graph has a fast decreasing gradient when the oscillator is first displaced until it reaches the x axis
- When the oscillator reaches the equilibrium position (x = 0), the graph is a horizontal line at x = 0 for the remaining time
describe a heavily damped oscillator
- When a heavily damped oscillator is displaced from the equilibrium, it will take a long time to return to its equilibrium position without oscillating
- The system returns to equilibrium more slowly than the critical damping case
- For example, door dampers are used on doors to prevent them slamming shut
Draw a displacement-time graph for a heavily damped oscillator
What are the key features of the displacement-time graph for a heavily damped oscillator
- There are no oscillations. This means the displacement does not pass zero
- The graph has a slow decreasing gradient from when the oscillator is first displaced until it reaches the x axis
- The oscillator reaches the equilibrium position (x = 0) after a long period of time, after which the graph remains a horizontal line for the remaining time
Free oscillation def
An oscillation where there are only internal forces (and no external forces) acting and there is no energy input
A free vibration always oscillates at its resonant frequency
forced oscillations def
Oscillations acted on by a periodic external force where energy is given in order to sustain oscillations
At what frequency are forced oscillations made to oscillate at
at the same frequency as the oscillator creating the external, periodic driving force
Driving frequency def
The frequency of forced oscillations
natural frequency (f0) def
the frequency of an oscillation when the oscillating system is allowed to oscillate freely
Resonance def
When the frequency of the applied force to an oscillating system is equal to its natural frequency, the amplitude of the resulting oscillations increases significantly
Why is resonance desirable, for example, for a child on a swing
energy is transferred from the driver to the oscillating system most efficiently
Therefore, at resonance, the system will be transferring the maximum kinetic energy possible
Draw the graph of driving frequency f against amplitude A of oscillations (resonance curve) AND describe its features
What are the effects of damping on resonance and decribe this effect on a resonance curve
- Damping reduces the amplitude of resonance vibrations
- The height and shape of the resonance curve will therefore change slightly depending on the degree of damping
Note: the natural frequency f0 of the oscillator will remain the same - As the degree of damping is increased, the resonance graph is altered in the following ways:
The amplitude of resonance vibrations decrease, meaning the peak of the curve lowers
The resonance peak broadens
The resonance peak moves slightly to the left of the natural frequency when heavily damped
- Therefore, damping reduced the sharpness of resonance and reduces the amplitude at resonant frequency
Draw the resonance curve, when damping occurs
Describe Barton’s pendulums
Under what condition does the time period for a simple pendulum equation work
For small amplitudes
Frequency of a simple pendulum in SHM
What kind of displacement-time graph does a SHM system produce
