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Physics A - Level > Further Mechanics and Thermal Physics (2): SHM > Flashcards

Further Mechanics and Thermal Physics (2): SHM Flashcards

1
Q

What is the definition of the equilibrium position for oscillating bodies?

A

The point at which the body will eventually come to a standstill

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2
Q

Describe the how the displacement of an oscillating object changes over one full cycle after being released form equilibrium?

A
  • 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
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3
Q

What is the definition of amplitude?

A

The maximum displacement of an oscillating object from its equilibrium position

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4
Q

If the amplitude is constant and there are no frictional forces, how are the oscillations described?

A

As free vibrations

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5
Q

What is the definition of Time Period, T?

A

The time taken for one complete cycle of oscillation (when the object passes through the same position in the same direction again)

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6
Q

What is the definition of the frequency of an oscillating object?

A

The number of full cycles per second made by an oscillating object

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7
Q

What is the equation for the phase difference between two oscillating objects in radians?

A

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

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8
Q

As an object moves away from equilibrium, how does its speed change?

A

Speed decreases

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9
Q

As an object moves towards equilibrium how does its speed change?

A

Speed increases

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10
Q

How does the displacement of an object that follows SHM change when it is released from a maximum displacement x?

A

The object oscillates between +x and -x following the motion of a cos wave or a sine wave (depends on the situation given)

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11
Q

Where does the force always act towards in SHM?

A

The equilibrium position

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12
Q

Where does acceleration always act towards in SHM?

A

The equilibrium position

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13
Q

How does the velocity of an object that follows SHM change when it is released from a +ve displacement x?

A

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
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14
Q

How does the acceleration if an object that follows SHM change when it is released from a +ve displacement x?

A

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
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15
Q

The gradient of what graph sows the variation of velocity with time?

A

displacement-time

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16
Q

The gradient of what graph shows the variation of acceleration with time?

A

velocity-time

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17
Q

In SHM in what way does the acceleration always act relative to the displacement?

A

The acceleration always acts in the opposite direction to the displacement

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18
Q

What is the definition of SHM?

A

The oscillating motion in which the acceleration is proportional to the displacement and always in the opposite direction to the displacement

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19
Q

What is the relationship between acceleration and the displacement in SHM?

A

a is directly proportional to -x

OR

a = -constant multiplied by displacement x

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20
Q

What does the constant of proportionality between acceleration and displacement in SHM depend on?

A

Depends on the time period of the oscillations

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21
Q

What does omega stand for in SHM?

A

The angular frequency

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22
Q

What is the equation for angular frequency of an object in SHM?

A

omega = 2(pi)/T

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23
Q

What is the equation for simple harmonic motion?

A

a = -(omega)^2x

where
x - displacement
omega - angular frequency

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24
Q

Does the time period of an object in SHM effect its amplitude and vice versa?

A

NO - the two values are independent of one another

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25
Q

What is maximum displacement equivalent to in SHM?

A

The amplitude of the oscillations

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26
Q

x(max) = …?

A

+/- ampltiude

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27
Q

When the maximum displacement = +A, what is the equation for SHM?

A

a = -(omega)^2A

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28
Q

When the maximum displacement = -A, what is the equation for SHM?

A

a = (omega)^2A

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29
Q

Does circular motion follow the same motion as SHM?

A

yes

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30
Q

How do you prove that the constant of proportionality for the acceleration of an SHM system is (omega)^2?

A

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

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31
Q

For the equation a = -(omega)^2x what does the variation of displacement with time depend on?

A

The initial displacement and initial velocity

-> displacement and velocity at t=0

32
Q

What is the equation for the displacement of an object in SHM at time t?

A

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
33
Q

What is the general equation for the displacement of an object following SHM at any particular time?

A

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
34
Q

V max = …?

A

omega x A

35
Q

a max = …?

A

(omega)^2 x A

36
Q

What is another name for the resultant force that acts towards the equilibrium in SHM?

A

The restoring force

37
Q

How can the frequency of oscillation of a loaded spring be changed? Why?

A
  • 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
38
Q

What piece of apparatus do you need to use when timing oscillations of a spring system?

A

A fiduciary marker placed at the centre of the oscillations

39
Q

Proof that a spring oscillates with SHM?

A

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

40
Q

What is the equation that links time period of a mass-spring system with the mass applied and spring constant of the spring?

A

T = 2(pi) root(m/k)

41
Q

Does the time period that a mass-spring system oscillate at depend on g?

A

No - a mass-spring system would oscillate with the same time period on the moon as the earth

42
Q

Proof that a simple pendulum oscillates with SHM?

A

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

43
Q

What is the equation that links time period of a simple pendulum with the L of the pendulum and g?

A

T = 2(pi) root(L/g)

where:
L - Length of the pendulum from the point of support to the centre of the bob

44
Q

Why is the equation T = 2(pi) root (L/g) valid for only small angles?

A

As for large angles you cannot approximate sin(theta) = s/L

45
Q

What are the only forces that act on a freely oscillating object?

A

Restoring force

46
Q

If friction was present on an oscillating system how would that affect the amplitude of the oscillations?

A

cause them to decrease

47
Q

What are the two stores of energy that are present in an oscillating system?

A

Kinetic energy
Potential energy

48
Q

For an oscillating system with no frictional forces acting on it what is the value of the total energy of the system?

A

Total energy constant and = Maximum potential energy

49
Q

What is the equation for the potential energy in an oscillating spring at any given point?

A

Ep = 1/2kx^2

50
Q

What is the equation for the total energy in an oscillating spring?

A

Et = 1/2kA^2

where
A - amplitude of oscillations

51
Q

What is the equation for the kinetic energy of an oscillating spring at any given point?

A

Et = Ek + Ep
Ek = Et - Ep = 1/2k(A^2-x^2)

52
Q

How do you prove the simple harmonic speed equation (find the speed at any given point)?

A

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

53
Q

Describe the shape of the lines for potential and kinetic energy on an energy-displacement graph?

A

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
54
Q

At x = 0, total energy =…?

A

Ek

55
Q

At +/- A, total energy =…?

A

Ep

56
Q

In real life scenarios what happens to a simple pendulum as time progresses? Why?

A

The amplitude of the oscillations decreases. This is because of friction/air resistance which slowly reduces the total energy of the system.

57
Q

What are the forces that reduce the total energy of an oscillating system called?

A

Dissipative forces

58
Q

What do dissipative forces do?

A

They dissipate the energy of the system to the surroundings as thermal energy

59
Q

If dissipative forces are present what is said to happen to the motion of the oscillations?

A

They are DAMPED

60
Q

What is light damping and when does it occur?

A

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

61
Q

What is critical damping an when does it occur?

A

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

62
Q

When is critical damping used?

A

Vehicle suspension systems

63
Q

What is heavy damping and when does it occur?

A

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

64
Q

What is an example of heavy damping?

A

A mass on a spring in thick oil

65
Q

What is the definition of periodic force?

A

When a force is applied at regular intervals to a pendulum/spring system

66
Q

What is the definition of natural frequency?

A

The frequency that a system oscillates at withpout a periodic force being applied

67
Q

When a periodic force is applied to an oscillating system the system undergoes…?

A

Forced vibrations/oscillations

68
Q

Describe how the amplitude of a systems oscillations changes as an applied frequency of increasing frequency is applied?

A

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

69
Q

What is the name of the frequency at which an object in SHM is at maximum amplitude of oscillations?

A

Resonant frequency

70
Q

At maximum amplitude what is the phase difference between the displacement at the periodic force?

A

1/2 (pi)

71
Q

At maximum amplitude what does the periodic force act in phase with?

A

The velocity of the object in SHM

72
Q

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?

A

No

73
Q

If the damping of the object in SHM is lighter, how will this effect the maximum amplitude of the oscillations at resonance?

A

Increase the max amplitude -> makes the resonance curve sharper

74
Q

What is the definition of resonance?

A

When the periodic force is exactly in phase with the velocity of the oscillating system

75
Q

If the damping of the object in SHM is lighter, how will this effect the value of the resonant frequency?

A

The resonant frequency will be closer to the natural frequency of the system

76
Q

For an oscillating system with little or no damping what is true at resonance?

A

The applied frequency of the periodic force = natural frequency of the system

77
Q

Describe how Bartons pendulums work?

A

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

Physics A - Level (24 decks)

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