Unit 4: Simple harmonic motion Flashcards
Describe one full cycle of motion
From maximum height at one side to maximum height on the other side and then back again
What is the lowest point of a swing referred to as?
The equilibrium position as this is where it will come to a standstill
Describe the motion of an oscillating object
An oscillating object moves repeatedly one way then in the opposite direction through its equilibrium position
What is displacement in terms of the equilibrium position?
The distance and direction from the equilibrium position
What is the amplitude of oscillations?
The maximum displacement of the oscillating object from equilibrium
What are features of free oscillations?
The amplitude is constant and there are no frictional forces
What is the time period?
The time for one complete cycle of oscillations
What is frequency and what is its unit?
The number of cycles per second and its unit is the hertz
What is the formula that relates time period and frequency?
T = 1/f
What is the formula for angular velocity in terms of the time period?
ω = 2π/T
What is the unit of angular velocity?
Radians per second
How can you tell if two objects are oscillating out of phase?
One object will reach maximum displacement on one side at a certain time, Δt, later than the other object
What is the formula for the phase difference in radians when two objects are oscillating at the same frequency and how is it different if you want the phase difference in degrees?
2πΔt/T in radians and 360Δt/T in degrees where Δt is the time between successive instants when the two objects are at maximum displacement in the same direction
Does the phase difference of two objects oscillating stay the same?
Yes
When do two objects oscillate in phase?
If Δt = T
What is a phase difference of 2π equal to?
0
How does the velocity of an oscillating object change in terms of the equilibrium?
An oscillating object speeds up as it returns to equilibrium and it slows down as it moves away from equilibrium
When will the amplitude of the oscillations be constant?
Provided friction is negligible
What does the gradient of a displacement-time graph represent?
Velocity
What does the gradient of a velocity-time graph represent?
Acceleration
When is the velocity of an object greatest?
When the gradient of the displacement-time graph is greatest i.e. when the object passes through equilibrium
When is the velocity zero?
When the gradient of the displacement-time graph is zero i.e. at maximum displacement
When is the acceleration greatest?
When the gradient of the velocity-time graph is greatest, this occurs when the velocity is zero and occurs at maximum displacement in the opposite direction
When is the acceleration zero?
When the gradient of the velocity-time graph is zero (the speed is at a maximum), this is when the displacement is zero
In which direction does acceleration act in terms of displacement?
The acceleration is always in the opposite direction to the displacement
What is simple harmonic motion defined as?
Oscillating motion in which the acceleration is proportional to the displacement and always in the opposite direction to the displacement
What does the constant of proportionality depend on in the simple harmonic equation and what produces a greater constant?
The time period of the oscillations and the constant is greater the shorter the time period as this means faster oscillations and therefore larger acceleration at any given displacement
What is the formula for simple harmonic motion?
a = -(ω^2)x where ω = 2π/T
What is the time period independent of?
The amplitude of the oscillations
What does the variation of displacement with time depend on ?
The initial displacement and the initial velocity
What is the SHM equation for the displacement when x = +A when t = 0 and the object has zero velocity at that instant?
x = Acos(2πft) where 2πft is in radians
What are SHM curves described as?
Sinusoidal
What is the general solution of a = -((2πf)^2)x?
x = Asin((2πft)+Φ) where Φ is the phase difference between the instants t = 0 and when x = 0
For any oscillating object, in which direction does the resultant force act?
The resultant force acting on the object acts towards the equilibrium position
What is the resultant force in SHM described as and why?
The restoring force as it always acts towards equilibrium
When is the acceleration (acting towards the equilibrium) proportional to the displacement
Provided the restoring force is proportional to the displacement from equilibrium
How can you test the oscillations of a mass-spring system?
Use two stretched springs and a trolley, when the trolley is displaced then released, it oscillates backwards and forwards, the first half-cycle of the trolley’s motion can be recorded using a length of ticker tape attached at one end to the trolley, when the trolley is released, the tape is pulled through a ticker timer that prints dots on the tape at a rate of 50 dots per second
What is done with the tape from the experiment of a mass-spring system?
A graph of displacement against time for the first half cycle can be drawn using the tape, the graph can be used to measure the time period which can be checked if the trolley mass m and the combined spring constant k are known
What can be used to record the oscillating motion of the trolley in the oscillating mass-spring system experiment?
A motion sensor linked to a computer
Why is the frequency of an oscillating spring reduced when the mass is increased?
The extra mass increases the inertia of the system, at a given displacement the object would therefore be slower than if the extra mass had not been added, each cycle of oscillation would therefore take longer
Why is the frequency of an oscillating spring reduced if weaker springs are used?
The restoring force on the trolley at any given displacement would be less so the object’s acceleration and speed at any given displacement would be less so each cycle of oscillation would therefore take longer
What is the formula for the time period of a mass spring system?
T = 2π(m/k)^1/2
What is ω in a mass spring system?
ω = (k/m)^1/2
Does the time period depend on g for a mass spring system?
No
What does the tension in a spring vary from?
It varies from mg + kA to mg - kA where A = the amplitude
When does the maximum tension in a spring occur?
When the spring is stretched as much as possible i.e. x = -A
When does the minimum tension in a spring occur?
When the spring is stretched as little as possible i.e. x = +A
What kind of graph should be produced when T^2 is plotted on the y axis against m on the x-axis?
A straight line through the origin with a gradient of 4π^2/k
What is the formula for the time period for a simple pendulum and in which cases does it apply?
T = 2π(l/g)^1/2 and it applies only if the angle of the thread to the vertical does not exceed about 10 degrees
What kind of graph should be produced when T^2 is plotted on the y axis against l on the x axis?
A straight line through the origin with a gradient of 4π^2/g
How can the time period of a pendulum be increased?
Increasing the length of l
What is the resultant force on a bob that is part of a simple pendulum at equilibrium?
T - mg = mv^2/l
How does a freely oscillating object oscillate and why does it oscillate in this way?
It oscillates with a constant amplitude because there is no friction acting on it - the only forces that act on it combine to form the restoring force
What would happen to the amplitude if friction was present?
The amplitude of the oscillations would gradually decrease and the oscillations would eventually cease
Describe the energy changes in a simple pendulum
The energy of the system changes from kinetic energy to potential energy and back again every half cycle after passing through equilibrium, provided friction is absent, the total energy of the system is constant and is equal to its maximum potential energy
What is the potential energy of a simple pendulum equal to?
0.5kx^2
What is the total energy of a simple pendulum equal?
0.5kA^2 where A is the amplitude of oscillations = kinetic energy + potential energy, this is the potential energy at maximum displacement which is the same as the kinetic energy at zero displacement
What is the kinetic energy of a simple pendulum equal to?
0.5k(A^2 - x^2)
At x = 0, what can the total energy be taken to equal?
Kinetic energy
What is the formula for v?
v = +/-ω(A^2-x^2)^1/2
What is the formula for vmax?
vmax = ωA
Describe the curves for potential energy, kinetic energy and total energy
The curve for potential energy is a parabola, for kinetic energy it is an inverted parabola and for the total energy it is a flat line
Why do the oscillations of a simple pendulum gradually die away?
Air resistance gradually reduces the total energy of the system
What are the forces causing the amplitude to decrease called?
Dissipative forces because they dissipate the energy of the system to the surroundings as thermal energy
What is said about the motion if dissipative forces are present?
The motion is damped
What is a result of the time period being independent of the amplitude when damping occurs?
Each cycle takes the same length of time as the oscillations die away
What is light damping?
The amplitude gradually decreases, reducing by the same fraction each cycle
What is critical damping?
It is just enough to stop the system oscillating after it has been displaced and released from equilibrium, the oscillating object returns to equilibrium in the shortest possible time without overshooting if the damping is critical
Why is critical damping important in mass-spring systems such as vehicle suspension systems?
An uncomfortable ride would be experienced if the damping was too light or too heavy
What is heavy damping?
The damping is so strong that the displaced object returns to equilibrium much more slowly than if the system is critically damped
What does the suspension system of a car include and what happens when the wheel is jolted?
A coiled spring near each wheel between the wheel axle and the car chassis and when the wheel is jolted, the spring smoothes out the force of the jolts
What prevents the chassis from bouncing up and down too much in a car suspension system and how?
An oil damper fitted with each spring, the flow of oil through valves in the piston of each damper provides a frictional force which damps the oscillating motion of the chassis
What would happen without oil dampers in a car suspension system?
The occupants of the car would continue to be thrown up and down until the oscillations died away
What are the dampers designed to ensure in a car suspension system?
The chassis returns to its equilibrium position in the shortest possible time after each jolt with little or no oscillations, the suspension system is therefore at/close to critical damping
What is a periodic force?
A force applied at regular intervals
What is natural frequency?
When a system oscillates without a periodic force being applied to it, its frequency is referred to as its natural frequency
When a periodic force is applied to an oscillating system, what does the response depend on?
The frequency of the periodic force
What does a system undergo when a periodic force is applied to a system?
Forced oscillations
How can a periodic force be applied to an oscillating system?
The equipment consists of a small object of fixed mass attached to two stretched springs, the bottom end of the lower spring is attached to a mechanical oscillator which is connected to a signal generator, the top end of the upper spring is fixed, the mechanical oscillator pulls repeatedly on the lower spring at a frequency that can be adjusted by adjusting the signal generator, the frequency of the oscillator is the applied frequency and the response of the system is measured from the amplitude of the oscillations of the object
What happens when the applied frequency approaches the natural frequency of the mass-spring system?
The amplitude of oscillations of the object increases more and more, the phase difference between the displacement and the periodic force increases from zero to π/2 at the natural frequency
What happens when the applied frequency is equal to the natural frequency of the mass-spring system?
The amplitude of the oscillations becomes very large (the lighter the damping in the system, the larger the amplitude becomes) and the phase difference between the displacement and the periodic force is π/2 at resonance, the periodic force is then exactly in phase with the velocity of the oscillating object
When is the system in resonance?
When the applied frequency is equal to the natural frequency
What happens as the applied frequency becomes increasingly larger than the natural frequency of the mass spring system?
The amplitude of the oscillations decreases more and more and the phase difference between the displacement and the periodic force increases from π/2 until the displacement is π radians out of phase with the periodic force
When is the amplitude of the oscillations greatest?
When the applied frequency is equal to the natural frequency provided the damping is light
Describe what happens at resonance in terms of amplitude and the periodic force
At resonance, the periodic force acts on the object at the same point in each cycle, causing the amplitude to increase to a maximum value limited only by damping, at maximum amplitude, energy supplied by the periodic force is lost at the same rate because of the effects of damping
When does resonance occur if the damping is not light?
A slightly lower frequency than the natural frequency, the lighter the damping, the closer the resonant frequency is to the natural frequency
When may a string be said to be in resonance?
When stationary waves are formed on a stretched spring
What does Barton’s pendulums involve?
Five simple pendulums, of different lengths hanging from a supporting thread which is stretched between two fixed points, a single driver pendulum of the same length as one of the other pendulums is also tied to the thread, the driver pendulum is displaced and released so it oscillates in a plane perpendicular to the plane of the pendulums at rest
What is the effect of the oscillating motion of the driver pendulum in Barton’s pendulums?
The oscillating motion of the driver pendulum is transmitted along the support thread, subjecting each of the other pendulums to forced oscillations
What happens to the pendulum that has the same length as the driver pendulum in Barton’s pendulums?
The pendulum that has the same length as the driver pendulum responds much more than any other pendulum as it has the same time period meaning they have the same natural frequency, this pendulum therefore oscillates in resonance with the driver pendulum because it is subjected to forced oscillations of the same frequency as its own natural frequency of oscillations
What does the resonance of the other pendulums in Barton’s pendulums depend on?
How close its length is to the length of the driver pendulum
Why can a bridge oscillate?
Due to its springiness and its mass
How can a bridge span be made to oscillate at resonance?
If it is not fitted with dampers, it can be made to oscillate at resonance if subjected to a suitable periodic force
How can a crosswind cause a periodic force on a bridge span?
Eddy currents created by the wind along the bridge span, if the wind speed is such that the periodic force is equal to the natural frequency of the bridge span, resonance can occur in the absence of damping
How is soldiers marching across a bridge related to resonance?
A steady trail of people in step with each other walking across a footbridge can cause resonant oscillations of the bridge span if there is insufficient damping, soldiers marching in columns are taught to break step when crossing a footbridge to avoid causing resonance
What makes the object exchange KE and PE?
The restoring force
Define what potential energy is for pendulums and what it is for springs
Gravitational PE for pendulums and elastic PE for masses on springs
What is mechanical energy and how does it vary?
The sum of the potential and kinetic energy and it stays constant as long as the motion is not damped
How far ahead is the velocity from the displacement?
A quarter of a cycle
Describe the velocity in terms of the centre of oscillation
It is positive if the object is moving away from the midpoint and negative if it’s moving towards the midpoint
What is an example of a simple harmonic oscillator?
A mass on a spring
What are free vibrations?
When there is no transfer of energy to or from the surroundings
If the driving frequency is much less than the natural frequency, what is their phase difference?
The two are in phase
If the driving frequency is much greater than the natural frequency, what is their phase difference?
The driver will be out of phase with the oscillator because the oscillator won’t be able to keep up
How does the plastic deformation of ductile materials reduce the amplitude of oscillations?
As the material changes shape, it absorbs energy so the oscillation will be smaller
Describe what the amplitude/driving frequency graph looks like for lightly damped systems
They have a very sharp resonance peak and their amplitude only increases dramatically when the driving frequency is very close to the natural frequency
Describe what the amplitude/driving frequency graph looks like for heavily damped systems
They have a flatter response, their amplitude doesn’t increase very much near the natural frequency and they aren’t as sensitive to the driving frequency
