Y2: Further Mechanics Flashcards
What is 1 radian
The angle when the arc length is equal to the radius
How do you convert from degrees to radians
x 2π/360
How do you convert from radians to degrees
x 360/2π
What is angular speed (ω)
The angle an object rotates through each second (rad/s)
ω = θ/t
What is tangential speed (v)
The linear speed of an object that is rotating around a point (m/s)
V = Arc length / t
What is the equation for angular speed
ω = V/r
V = arc length / t
∴ V = θr/t
∴ V/r = θ/t
∴ ω = V/r
What is frequency in circular motion
The number of complete revolutions per second (Hz)
f = 1/T
What is the equation linking angular speed and frequency
ω = 2πf
ω = θ/t
∴ ω = 2π/T
∴ ω = 2πf
What is centripetal acceleration
Acceleration towards the centre of rotation
What is the equation for centripetal acceleration
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
What is the centripetal force
The force that causes the centripetal acceleration (N)
What is the equation for centripetal force
F = m(V^2)/r = m(ω^2)r
F = ma
∴F = m(V^2)/r
What is simple harmonic motion
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
What is displacement in SHM (x)
The distance the object is from the equilibrium position
- Varies as a sine/cosine wave on a displacement-time graph
- Max value = Amplitude (A)
What is the velocity in SHM
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
What is acceleration in SHM
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
What is the phase difference of two waves
A measure of how much one wave lags behind another
(measured in radians, degrees or fractions of a circle)
What is the phase difference of 2 waves in phase
0 or 2π radians
What is the phase difference of 2 waves out of phase
π radians
What is the relationship between velocity and displacement in SHM
Velocity is π/2 radians out of phase with displacement
What is the relationship between acceleration and velocity in SHM
Acceleration is π/2 radians out of phase with velocity
What is the relationship between acceleration and displacement in SHM
Acceleration is in anti-phase with displacement (π radians out of phase)
∴ a ∝ -x
What is frequency in SHM
The number of cycles per second
f = 1/T
What is the Period of SHM
Time taken to complete one cycle
T = 1/f
How does energy change during SHM
As an object oscillates, the potential and kinetic energies exchange.
∴ Ep + Ek = Mechanical energy (constant if no damping)
How does kinetic energy change in relation to displacement in SHM
Max kinetic energy when x=0
0 kinetic energy when x=A
How does potential energy (GPE/EPE) change in relation to displacement in SHM
Max potential energy when x=A
0 potential energy when x=0
When considering SHM as a projection of circular motion, what is the equation for displacement
x = Acos(ωt)
x = Horizontal component of position
∴ x = rcosθ
r = Max displacement
∴ r = A
ω = θ/t
∴ θ = ωt
∴ x = Acos(ωt)
When considering SHM as a projection of circular motion, what is the equation for acceleration
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
What is the restoring force of a spring
F = -kx
Hooke’s law: F = kx
Negative, as acting in the opposite direction
How does angular speed of SHM relate to the spring constant
ω^2 = k/m
Restoring force = -kx
F = ma
∴ ma = -kx
∴ -a/x = k/m
a = -(ω^2)x
∴ ω^2 = k/m
When considering SHM as a projection of circular motion, what is the equation for velocity
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)
What is the equation for the period of oscillation for a mass on a spring
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)
How will altering the mass of a mass-spring oscillator affect the time period
Increasing the mass will increase T^2, as √m ∝ T
How will altering the Spring constant of a mass-spring oscillator affect the time period
Increasing k will decrease T^2, as √(1/k) ∝ T
How will altering the Amplitude of a mass-spring oscillator affect the time period
Changing A will not alter the time period, as T is constant for all displacements
What is the equation for the period of oscillation for a simple pendulum
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)
How will altering the Length of a pendulum affect the time period
Increasing the length will increase T^2, as √l ∝ T
How will altering the mass of a pendulum affect the time period
Changing the mass will not alter the time period, as T is constant for masses
How will altering the Amplitude of a pendulum affect the time period
Changing A will not alter the time period, as T is constant for all displacements
What are free vibrations
Oscillations without energy transfer to or from the surroundings
(Not possible in practice, as x remains constant)
What is the frequency of a free vibration
Resonant/natural frequency
What are forced vibrations
Oscillations with an external driving force
What is the driving frequency
The frequency of the external, periodic force
What happens if the driving freq. < Natural freq.
The 2 oscillations are in phase, as the oscillator follows the driver
What happens if the driving freq. > Natural freq.
The oscillator and driver end up in antiphase
What is resonance
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)
What happens as the driving frequency increases
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)
What is damping
When dissipative forces are present, so energy is lost to the surroundings
What is light damping
Damping that causes the amplitude to gradually decrease by the same fraction each cycle
eg. Air resistance on a pendulum
What is heavy damping
Damping where no oscillation occurs, and the mass returns slowly to equilibrium
eg. Heavy doors to prevent slamming
What is critical damping
Damping that is just enough so the mass returns to the equilibrium in the shortest possible time (No oscillations)
e. Car suspension
What is overdamping
Damping that causes the system to take longer to return to the equilibrium that critical damping
How does damping effect the amplitude in relation to the driving frequency
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.
