Y2: Further Mechanics Flashcards

Question

How does energy change during SHM

Answer 1

As an object oscillates, the potential and kinetic energies exchange. ∴ Ep + Ek = Mechanical energy (constant if no damping)

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

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)

Answer 9

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)

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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)

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

Resonant/natural frequency

Answer 19

Oscillations with an external driving force

Answer 20

The frequency of the external, periodic force

Answer 21

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

Answer 22

The oscillator and driver end up in antiphase

Answer 23

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)

Answer 24

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)

Answer 25

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

Answer 26

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

Answer 27

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

Answer 28

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

Answer 29

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

Answer 30

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.