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Physics > Y2: Further Mechanics > Flashcards

Y2: Further Mechanics Flashcards

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1
Q

What is 1 radian

A

The angle when the arc length is equal to the radius

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2
Q

How do you convert from degrees to radians

A

x 2π/360

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3
Q

How do you convert from radians to degrees

A

x 360/2π

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4
Q

What is angular speed (ω)

A

The angle an object rotates through each second (rad/s)

ω = θ/t

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5
Q

What is tangential speed (v)

A

The linear speed of an object that is rotating around a point (m/s)

V = Arc length / t

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6
Q

What is the equation for angular speed

A

ω = V/r

V = arc length / t
∴ V = θr/t
∴ V/r = θ/t
∴ ω = V/r

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7
Q

What is frequency in circular motion

A

The number of complete revolutions per second (Hz)

f = 1/T

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8
Q

What is the equation linking angular speed and frequency

A

ω = 2πf

ω = θ/t
∴ ω = 2π/T
∴ ω = 2πf

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9
Q

What is centripetal acceleration

A

Acceleration towards the centre of rotation

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10
Q

What is the equation for centripetal acceleration

A

a = (V^2)/r = (ω^2)r

Arc length = rθ = VΔt
Arc length/r = ΔV/V (same proportions)
∴ VΔt/r = ΔV/V
∴ (V^2)/r = ΔV/Δt = acceleration
∴ a = (V^2)/r

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11
Q

What is the centripetal force

A

The force that causes the centripetal acceleration (N)

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12
Q

What is the equation for centripetal force

A

F = m(V^2)/r = m(ω^2)r

F = ma
∴F = m(V^2)/r

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13
Q

What is simple harmonic motion

A

An oscillation in which the acceleration of an object is directly proportional to it’s displacement from it’s equilibrium position, and is directed towards the equilibrium

∴ a ∝ -x

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14
Q

What is displacement in SHM (x)

A

The distance the object is from the equilibrium position
- Varies as a sine/cosine wave on a displacement-time graph
- Max value = Amplitude (A)

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15
Q

What is the velocity in SHM

A

Speed of the oscillating object in a +/- direction
- The Gradient of a displacement-time graph
- Max value = Max gradient of x-t graph (x=0)
- Max value = ωA

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16
Q

What is acceleration in SHM

A

The rate of change of speed for an oscillating object
- The Gradient of a velocity-time graph
- Max value = Max gradient of v-t graph (v=0)
- Max value = (ω^2)A

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17
Q

What is the phase difference of two waves

A

A measure of how much one wave lags behind another
(measured in radians, degrees or fractions of a circle)

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18
Q

What is the phase difference of 2 waves in phase

A

0 or 2π radians

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19
Q

What is the phase difference of 2 waves out of phase

A

π radians

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20
Q

What is the relationship between velocity and displacement in SHM

A

Velocity is π/2 radians out of phase with displacement

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21
Q

What is the relationship between acceleration and velocity in SHM

A

Acceleration is π/2 radians out of phase with velocity

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22
Q

What is the relationship between acceleration and displacement in SHM

A

Acceleration is in anti-phase with displacement (π radians out of phase)

∴ a ∝ -x

23
Q

What is frequency in SHM

A

The number of cycles per second
f = 1/T

24
Q

What is the Period of SHM

A

Time taken to complete one cycle
T = 1/f

25
Q

How does energy change during SHM

A

As an object oscillates, the potential and kinetic energies exchange.

∴ Ep + Ek = Mechanical energy (constant if no damping)

26
Q

How does kinetic energy change in relation to displacement in SHM

A

Max kinetic energy when x=0
0 kinetic energy when x=A

27
Q

How does potential energy (GPE/EPE) change in relation to displacement in SHM

A

Max potential energy when x=A
0 potential energy when x=0

28
Q

When considering SHM as a projection of circular motion, what is the equation for displacement

A

x = Acos(ωt)

x = Horizontal component of position
∴ x = rcosθ
r = Max displacement
∴ r = A
ω = θ/t
∴ θ = ωt
∴ x = Acos(ωt)

29
Q

When considering SHM as a projection of circular motion, what is the equation for acceleration

A

a = -(ω^2)x

a=(ω^2)r
∴ in horizontal plane, a=(ω^2)rcosθ
x = rcosθ
∴ a= -(ω^2)rcosθ
Negative, as occurs in opposite direction

30
Q

What is the restoring force of a spring

A

F = -kx

Hooke’s law: F = kx
Negative, as acting in the opposite direction

31
Q

How does angular speed of SHM relate to the spring constant

A

ω^2 = k/m

Restoring force = -kx
F = ma
∴ ma = -kx
∴ -a/x = k/m
a = -(ω^2)x
∴ ω^2 = k/m

32
Q

When considering SHM as a projection of circular motion, what is the equation for velocity

A

V = ±ω√(A^2-x^2)

For a spring:
Ep = (1/2)kx^2
Etotal = (1/2)kA^2
∴ Ek = (1/2)k(A^2-x^2)
∴ (1/2)mV^2 = (1/2)k(A^2-x^2)
∴ V^2 = (ω^2)(A^2-x^2)
∴ V = ±ω√(A^2-x^2)

33
Q

What is the equation for the period of oscillation for a mass on a spring

A

T = 2π√(m/k)

Restoring force = -kx
F = ma
a = -(ω^2)x
∴ -(2πf)^2(mx) = -kx
∴ f = (1/2π)√(k/m)
T=1/f
∴ T = 2π√(m/k)

34
Q

How will altering the mass of a mass-spring oscillator affect the time period

A

Increasing the mass will increase T^2, as √m ∝ T

35
Q

How will altering the Spring constant of a mass-spring oscillator affect the time period

A

Increasing k will decrease T^2, as √(1/k) ∝ T

36
Q

How will altering the Amplitude of a mass-spring oscillator affect the time period

A

Changing A will not alter the time period, as T is constant for all displacements

37
Q

What is the equation for the period of oscillation for a simple pendulum

A

T = 2π√(l/g)

Restoring force = Vertical component of mg
∴ F = -(mg)Sinθ
F = ma
∴ ma = (mg)Sinθ
∴ a = -gSinθ
For small angles, sinθ ≈ θ
∴ a = -gθ
Arc length = s
∴ s = rθ
∴θ = s/r
∴ a = -g(s/r)
a = -(ω^2)x =((2πf)^2)x
For a pendulum, s=x
∴ -g(x/l) =-((2πf)^2)x
∴ f = (1/2π)√(g/l)
T=1/f
∴ T = 2π√(l/g)

38
Q

How will altering the Length of a pendulum affect the time period

A

Increasing the length will increase T^2, as √l ∝ T

39
Q

How will altering the mass of a pendulum affect the time period

A

Changing the mass will not alter the time period, as T is constant for masses

40
Q

How will altering the Amplitude of a pendulum affect the time period

A

Changing A will not alter the time period, as T is constant for all displacements

41
Q

What are free vibrations

A

Oscillations without energy transfer to or from the surroundings
(Not possible in practice, as x remains constant)

42
Q

What is the frequency of a free vibration

A

Resonant/natural frequency

43
Q

What are forced vibrations

A

Oscillations with an external driving force

44
Q

What is the driving frequency

A

The frequency of the external, periodic force

45
Q

What happens if the driving freq. < Natural freq.

A

The 2 oscillations are in phase, as the oscillator follows the driver

46
Q

What happens if the driving freq. > Natural freq.

A

The oscillator and driver end up in antiphase

47
Q

What is resonance

A

When the frequency of the driving force is equal to the natural frequency, causing the maximum amplitude of the oscillator (if there is no damping)

48
Q

What happens as the driving frequency increases

A

As the driving freq. approaches the natural freq., the system is resonating, with the maximum amplitude increasing.
Beyond the natural freq., the maximum amplitude decreases
(only if no damping)

49
Q

What is damping

A

When dissipative forces are present, so energy is lost to the surroundings

50
Q

What is light damping

A

Damping that causes the amplitude to gradually decrease by the same fraction each cycle
eg. Air resistance on a pendulum

51
Q

What is heavy damping

A

Damping where no oscillation occurs, and the mass returns slowly to equilibrium
eg. Heavy doors to prevent slamming

52
Q

What is critical damping

A

Damping that is just enough so the mass returns to the equilibrium in the shortest possible time (No oscillations)
e. Car suspension

53
Q

What is overdamping

A

Damping that causes the system to take longer to return to the equilibrium that critical damping

54
Q

How does damping effect the amplitude in relation to the driving frequency

A

More damping causes a flatter response, as the max amplitude at the natural frequency decreases, will a less sharp increase at this point when more damping is present.

Physics (14 decks)

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