Further mechanics - Simple harmonic motion

Question 1

What is the necessary condition for simple harmonic motion (SHM)?

Answer 1

Acceleration of a particle proportional too displacement in the opposite direction

a=-kx

Question 2

In the equation denoting simple harmonic motion as a=-kx, what is the constant k?

Answer 2

ω² ~ angular velocity²

Question 3

Where in the cycle is the particle’s velocity a maximum?

Answer 3

The centre of its oscillation (Velocity = 0 at maximum displacement)

Question 4

Where in the cycle is the particle’s Acceleration a maximum?

Answer 4

When the displacement of the particle is at a maximum (acceleration = 0 at centre of oscillation)

Question 5

What is the relationship between acceleration and displacement?

Answer 5

Acceleration is the rate of change of the rate of change (velocity) of displacement

Question 6

How do you calculate the maximum velocity?

Answer 6

Vmax = ω.A

Question 7

What trig function generally maps the graph of displacement /time ?

Answer 7

Cosine

Question 8

What equation (hint: graphical) links displacement, amplitude, angular velocity and time?

Answer 8

x = A cos(ωt)

Question 9

How do you interpret amplitude and time period from a displacement time graph?

Answer 9

Amplitude = Maximum displacement (if cosine, y intercept)

Time period = period of curve

Question 10

What equation (hint: graphical) links velocity, amplitude, angular velocity and time?

Answer 10

V = -ω.A.Sin(ωt)

Question 11

What equation (hint: graphical) links acceleration, amplitude, angular velocity and time?

Answer 11

a = -ω².A.Cos(ωt)

Question 12

What is the relationship between acceleration, velocity and displacement?

Answer 12

da/dt = v
dv/dt = x

Question 13

What trig function generally maps the graph of velocity /time ?

Answer 13

-Sine (NEGATIVE)

Question 14

What trig function generally maps the graph of acceleration /time ?

Answer 14

-Cosine (NEGATIVE)

Question 15

How is the total energy against time represented on a graph in an ideal simple harmonic system?

Answer 15

A straight line through the y axis
Ep+Ek = constant

Question 16

How do you derive an equation for kinetic energy equation using SHM equations?

Answer 16

Ek = 1/2.m.v²
V = -ω.A.Sin(ωt)
1/2.m.ω².A².Sin²(ωt)

Question 17

How do you derive an equation for maximum kinetic energy equation using SHM equations?

Answer 17

Ek = 1/2.m.v²
Vmax = ω.A
1/2.m.ω².A²

Question 18

On an energy against time graph, describe the curves of Ep and Ek.

Answer 18

Ek = Inverse parabola
Ep = Parabola (x²)

Question 19

Why can a mass spring system undergo SHM? Can you derive it?

Answer 19

The acceleration is directly proportional to the negative displacement (opposite direction).

F=ma F=-kx (k is the spring constant)
a=-k/m . x
a = omega^2. A A is interchangable with x
omega = 2pi / T and so on…

Question 20

Where in the cycle is the particle’s kinetic energy a maximum?

Answer 20

Centre/ origin of oscillations

Question 21

Where in the cycle is a particle’s potential energy a maximum?

Answer 21

Maximum displacement of the oscillation

Question 22

What are the limitations of a vertical spring system?

Answer 22

Must be obeying hooks law (elastic region) and there is already a slight extension from the springs natural length when applying the mass when it is in equilibrium; use relatively low masses. Use small amplitudes so don’t over stretch spring.

Question 23

What type of spring system can the time period equation T =2π √(m/k) ?

Answer 23

Horizontal with no friction

Question 24

For a simple pendulum, how do you calculate the angular velocity?

Answer 24

ω=√(g/l)

Question 25

For a simple pendulum, how do you calculate the time period?

Answer 25

T=2π√(l/g)

Question 26

Why does a simple pendulum undergo SHM?

Answer 26

By drawing a FBD it can be seen that : mgSinθ = -ma (tangential acceleration) a = -gSinθ Due too small angle approximation, Sin θ ≈ θ ᵣ (radians) a = -gθ and x=lθ so a= -(g/l) . x g and l are constant therefore a ∝ -x

Question 27

For a horizontal spring system with multiple springs, how do you calculate the Time period and Angular velocity?

Answer 27

Original equations but add the total spring constants of the multiple springs. T=2π√(m/k1+k2) ω=√(k1+k2/m)

Question 28

For a horizontal spring system how do you calculate the Time period and Angular velocity?

Answer 28

T=2π√(m/k) ω=√(k/m)

Question 29

What is damping?

Answer 29

The reduction in energy and amplitude of oscillations due to resistive forces on the oscillating system.

Question 30

What happens too the time period during light dampening?

Answer 30

Nothing, it remains constant

Question 31

What happens to the amplitude of oscillations during light dampening?

Answer 31

Decays exponentially with time

Question 32

Describe the forces acting on an oscillator during light dampening

Answer 32

Resistive forces oppose the motion, Restorative forces bring the oscillator back to equilibrium

Question 33

What is critical damping?

Answer 33

The oscillator returns to rest in the shortest time possible, without oscillating.

Question 34

What is heavy damping?

Answer 34

The oscillator return to equilibrium without any oscillations over a longer period of time (over damping).

Question 35

What is the driving frequency?

Answer 35

Frequency of forced oscillations (By a driving force)

Question 36

What is the natural frequency?

Answer 36

The frequency of oscillation when the system is allowed to oscillate freely.

Question 37

When does the resonance peak occur?

Answer 37

Natural frequency = Driving frequency

Question 38

What happens at the resonance peak?

Answer 38

Amplitude increases significantly due to the most efficient energy transfer between Ek and Ep

Question 39

What effect does dampening have on the natural frequency of a system?

Answer 39

None!

Question 40

What effect does dampening have on the Amplitude of resonance?

Answer 40

Reduces the Amplitude

Question 41

What effect does dampening have on the resonant frequency of a SHSystem ?

Answer 41

The resonant frequency becomes < Natural frequency On an A/f graph, shit too the left and less steep peak

Question 42

As the degree of damping increases, what happens to the resonance peak?

Answer 42

The peak of the curve lowers and it becomes broader.Furthermore, the peak of resonance itself moves further to the left.