Further Mechanics and Thermal Physics (2): SHM Flashcards
What is the definition of the equilibrium position for oscillating bodies?
The point at which the body will eventually come to a standstill
Describe the how the displacement of an oscillating object changes over one full cycle after being released form equilibrium?
- Displacement decreases as it returns to equilibrium
- Displacement reverses and increases as it moves away from the equilibrium in the opposite direction
- Displacement decreases again as it returns to equilibrium
- Displacement increases as it moves away from equilibrium towards its starting position
What is the definition of amplitude?
The maximum displacement of an oscillating object from its equilibrium position
If the amplitude is constant and there are no frictional forces, how are the oscillations described?
As free vibrations
What is the definition of Time Period, T?
The time taken for one complete cycle of oscillation (when the object passes through the same position in the same direction again)
What is the definition of the frequency of an oscillating object?
The number of full cycles per second made by an oscillating object
What is the equation for the phase difference between two oscillating objects in radians?
phase difference in radians = 2(pi)(delta)t/T
where:
(delta)t is the time between successive instants when the two objects are at maximum displacement in the same direction
As an object moves away from equilibrium, how does its speed change?
Speed decreases
As an object moves towards equilibrium how does its speed change?
Speed increases
How does the displacement of an object that follows SHM change when it is released from a maximum displacement x?
The object oscillates between +x and -x following the motion of a cos wave or a sine wave (depends on the situation given)
Where does the force always act towards in SHM?
The equilibrium position
Where does acceleration always act towards in SHM?
The equilibrium position
How does the velocity of an object that follows SHM change when it is released from a +ve displacement x?
The velocity follows the motion of a -ve sine wave (the gradient at each point of a displacement-time graph)
- Initially, the object travels away from the +ve displacement point, therefore initial velocity is -VE
- The object reaches a peak, maximum negative velocity point, as it goes past the equilibrium position
- The objects velocity becomes less negative as it reaches the -ve maximum displacement on the other side (this is when the graph goes through 0)
- The velocity then becomes increasingly positive as the object oscillates back down to the equilibrium position and reaches a second peak which is the same magnitude as the first but +ve
- The velocity then decreases down to zero as it returns to its initial displacement position
How does the acceleration if an object that follows SHM change when it is released from a +ve displacement x?
The acceleration follows the motion of a -ve cos wave (or the gradient at each point of a velocity-time graph OR opposite to the graph of displacement-time)
- Initially, acceleration is -ve as you have a positive displacement
- The acceleration decreases to 0 when as the object passes through its equilibrium position as the velocity is at a maximum
- The acceleration increases in the positive direction (acts in the positive direction) as the object gains -ve displacement
- Acceleration reaches a maximum at the maximum -ve displacement then decreases back down to 0 as the object passes through the equilibrium position again
- Finally, the acceleration increases in the -ve direction a the object returns to its original displacement
The gradient of what graph sows the variation of velocity with time?
displacement-time
The gradient of what graph shows the variation of acceleration with time?
velocity-time
In SHM in what way does the acceleration always act relative to the displacement?
The acceleration always acts in the opposite direction to the displacement
What is the definition of SHM?
The oscillating motion in which the acceleration is proportional to the displacement and always in the opposite direction to the displacement
What is the relationship between acceleration and the displacement in SHM?
a is directly proportional to -x
OR
a = -constant multiplied by displacement x
What does the constant of proportionality between acceleration and displacement in SHM depend on?
Depends on the time period of the oscillations
What does omega stand for in SHM?
The angular frequency
What is the equation for angular frequency of an object in SHM?
omega = 2(pi)/T
What is the equation for simple harmonic motion?
a = -(omega)^2x
where
x - displacement
omega - angular frequency
Does the time period of an object in SHM effect its amplitude and vice versa?
NO - the two values are independent of one another
What is maximum displacement equivalent to in SHM?
The amplitude of the oscillations
x(max) = …?
+/- ampltiude
When the maximum displacement = +A, what is the equation for SHM?
a = -(omega)^2A
When the maximum displacement = -A, what is the equation for SHM?
a = (omega)^2A
Does circular motion follow the same motion as SHM?
yes
How do you prove that the constant of proportionality for the acceleration of an SHM system is (omega)^2?
Using a set up where you compare the circular motion of a ball with SHM of a pendulum -> have a projector which shows that the shadows of both of the items remain in phase
- Start the two, ball and bob, in the same position with P having the coordinates (rcos(theta), rsin(theta))
- They will follow the same motion
Of ball on turntable
a = -v^2/r
as minus sign indicates direction is towards the centre of the circle
a = -(omega)^2r
component of acceleration parallel to the screen ->
ax = acos(theta)
ax = -(omega)^2rcos(theta)
as x = rcos(theta)
ax = -(omega)^2x
For the equation a = -(omega)^2x what does the variation of displacement with time depend on?
The initial displacement and initial velocity
-> displacement and velocity at t=0
What is the equation for the displacement of an object in SHM at time t?
x = Acos(omega x t)
A - amplitude
omega - angular frequency (2(pi)/T)
- proved by the fact that at t = 0, cos(omega x t) = 1
therefore x = A
What is the general equation for the displacement of an object following SHM at any particular time?
x = Acos(omega(t) + theta)
where
A - amplitude
Theta - phase difference between when t = 0 and when x = 0
- The other equation is only true for if the SHM starts from when the oscillations start from the centre moving +ve
V max = …?
omega x A
a max = …?
(omega)^2 x A
What is another name for the resultant force that acts towards the equilibrium in SHM?
The restoring force
How can the frequency of oscillation of a loaded spring be changed? Why?
- Increasing the mass of the load -> This is because the extra mass will increase the inertia of the system . At a given displacement the load would therefore be moving slower -> each cycle would take LONGER
- Decreasing the spring constant/stiffness constant -> The restoring force on the load at any given displacement would be less, therefore the trolley’s acceleration and speed at any given displacement would be less -> each cycle would take LONGER
What piece of apparatus do you need to use when timing oscillations of a spring system?
A fiduciary marker placed at the centre of the oscillations
Proof that a spring oscillates with SHM?
Consider a small mass, m attached to spring:
Assuming the spring obeys Hooke’s Law T (tension in the spring) = k(delta)L
When the spring is oscillating and has a displacement x from the equilibrium, the change in tension provides the restoring force on the object
Using T = k(delta)L -> T = -kx, where the negative sign represents the fact that the change in tension always tries to restore the object back to its equilibrium position
Restoring force = -kx
a = restoring force/mass = -kx/m
this can be written in the form a = -(omega)^2x
where omega = k/m
What is the equation that links time period of a mass-spring system with the mass applied and spring constant of the spring?
T = 2(pi) root(m/k)
Does the time period that a mass-spring system oscillate at depend on g?
No - a mass-spring system would oscillate with the same time period on the moon as the earth
Proof that a simple pendulum oscillates with SHM?
Consider a simple pendulum with a bob of mass m and a thread of length, L:
If the bob is displaced from its equilibrium and released it oscillates about the point of equilibrium
At displacement s from the lowest point when the thread is at angle theta to the vertical, the weight mg has components
-> mgcos(theta) perpendicular to the path of the bob
-> mgsin(theta) along the path of the bob
Therefore, the restoring force = -mgsin(theta)
-> acceleration = -gsin(theta)
As long as theta does not exceed approximately 10 degrees sin(theta) = s/L
-> a = (-g/L)s = -(omega)^2s where omega^2 = g/L
What is the equation that links time period of a simple pendulum with the L of the pendulum and g?
T = 2(pi) root(L/g)
where:
L - Length of the pendulum from the point of support to the centre of the bob
Why is the equation T = 2(pi) root (L/g) valid for only small angles?
As for large angles you cannot approximate sin(theta) = s/L
What are the only forces that act on a freely oscillating object?
Restoring force
If friction was present on an oscillating system how would that affect the amplitude of the oscillations?
cause them to decrease
What are the two stores of energy that are present in an oscillating system?
Kinetic energy
Potential energy
For an oscillating system with no frictional forces acting on it what is the value of the total energy of the system?
Total energy constant and = Maximum potential energy
What is the equation for the potential energy in an oscillating spring at any given point?
Ep = 1/2kx^2
What is the equation for the total energy in an oscillating spring?
Et = 1/2kA^2
where
A - amplitude of oscillations
What is the equation for the kinetic energy of an oscillating spring at any given point?
Et = Ek + Ep
Ek = Et - Ep = 1/2k(A^2-x^2)
How do you prove the simple harmonic speed equation (find the speed at any given point)?
Ek = 1/2mv^2
-> 1/2mv^2 = 1/2k(A^2-x^2)
as omega^2 = k/m
v^2 = omega^2(A^2-x^2)
Therefore,
v = +/- omega^2(A^2-x^2)
NB - making displacement = 0 gives the maximum speed = omega x A
Describe the shape of the lines for potential and kinetic energy on an energy-displacement graph?
Y axis - Energy
X axis - Displacement from -A to +A
- The line of Ep is a positive parabola from max at -A down to 0 at 0 and then back up to max at +A
- The line of Ek is an inverted parabola of Ep
At x = 0, total energy =…?
Ek
At +/- A, total energy =…?
Ep
In real life scenarios what happens to a simple pendulum as time progresses? Why?
The amplitude of the oscillations decreases. This is because of friction/air resistance which slowly reduces the total energy of the system.
What are the forces that reduce the total energy of an oscillating system called?
Dissipative forces
What do dissipative forces do?
They dissipate the energy of the system to the surroundings as thermal energy
If dissipative forces are present what is said to happen to the motion of the oscillations?
They are DAMPED
What is light damping and when does it occur?
Light damping - when the amplitude of an oscillation decreases, reducing by the same fraction each time
-> Occurs when the time period is independent of the amplitude so each cycle takes the same length of time for the oscillations to die away
What is critical damping an when does it occur?
Critical damping - when the oscillating system returns to equilibrium in the shortest possible time without overshooting
-> Just enough to stop the system from oscillating after it has been displaced from equilibrium and released
-> A straight line is drawn from the displacement with a very steep gradient down to 0 displacement and then straight across as time progresses on a displacement-time graph
When is critical damping used?
Vehicle suspension systems
What is heavy damping and when does it occur?
Heavy damping - Causes no oscillating to occur and occurs when the damping is so strong that the displaced object returns to equilibrium much more slowly that if the system was critically damped
-> A straight line is drawn from the displacement with a shallow gradient downwards
What is an example of heavy damping?
A mass on a spring in thick oil
What is the definition of periodic force?
When a force is applied at regular intervals to a pendulum/spring system
What is the definition of natural frequency?
The frequency that a system oscillates at withpout a periodic force being applied
When a periodic force is applied to an oscillating system the system undergoes…?
Forced vibrations/oscillations
Describe how the amplitude of a systems oscillations changes as an applied frequency of increasing frequency is applied?
Amplitude of oscillations of the system increases until they reach a maximum amplitude which is when the phase difference between between the displacement and periodic force increases from 0 to 1/2(pi)
After that the oscillations decrease from the maximum amplitude
What is the name of the frequency at which an object in SHM is at maximum amplitude of oscillations?
Resonant frequency
At maximum amplitude what is the phase difference between the displacement at the periodic force?
1/2 (pi)
At maximum amplitude what does the periodic force act in phase with?
The velocity of the object in SHM
Does the periodic force have to be exactly 1/2 (pi) out of phase in order to cause an increase in the amplitude of the oscillations?
No
If the damping of the object in SHM is lighter, how will this effect the maximum amplitude of the oscillations at resonance?
Increase the max amplitude -> makes the resonance curve sharper
What is the definition of resonance?
When the periodic force is exactly in phase with the velocity of the oscillating system
If the damping of the object in SHM is lighter, how will this effect the value of the resonant frequency?
The resonant frequency will be closer to the natural frequency of the system
For an oscillating system with little or no damping what is true at resonance?
The applied frequency of the periodic force = natural frequency of the system
Describe how Bartons pendulums work?
Pendulums D, P, Q, R, S, T of different lengths are all hanging from a supportive thread stretched between two fixed points
- A driver pendulum D is of equal lengths to one of the other pendulums and is displaced and released so that it oscillates in a plane perpendicular to the planes of the pendulums at rest
- The effect of the oscillating motion of D is transmitted along the support thread subject the other pendulums to FORCED OSCILLATIONS
- The pendulum that has the same length as D will be subject to the greatest amplitude oscillations as it has the same length and therefore, the same period as D and will have the same natural frequency as D
- Therefore, the pendulum with the same natural frequency acts in resonance with D as it is subjected to forced oscillations of the same frequency of its own natural frequency
