Further mechanics - Simple harmonic motion Flashcards
What is the necessary condition for simple harmonic motion (SHM)?
Acceleration of a particle proportional too displacement in the opposite direction
a=-kx
In the equation denoting simple harmonic motion as a=-kx, what is the constant k?
ω² ~ angular velocity²
Where in the cycle is the particle’s velocity a maximum?
The centre of its oscillation (Velocity = 0 at maximum displacement)
Where in the cycle is the particle’s Acceleration a maximum?
When the displacement of the particle is at a maximum (acceleration = 0 at centre of oscillation)
What is the relationship between acceleration and displacement?
Acceleration is the rate of change of the rate of change (velocity) of displacement
How do you calculate the maximum velocity?
Vmax = ω.A
What trig function generally maps the graph of displacement /time ?
Cosine
What equation (hint: graphical) links displacement, amplitude, angular velocity and time?
x = A cos(ωt)
How do you interpret amplitude and time period from a displacement time graph?
Amplitude = Maximum displacement (if cosine, y intercept)
Time period = period of curve
What equation (hint: graphical) links velocity, amplitude, angular velocity and time?
V = -ω.A.Sin(ωt)
What equation (hint: graphical) links acceleration, amplitude, angular velocity and time?
a = -ω².A.Cos(ωt)
What is the relationship between acceleration, velocity and displacement?
da/dt = v
dv/dt = x
What trig function generally maps the graph of velocity /time ?
-Sine (NEGATIVE)
What trig function generally maps the graph of acceleration /time ?
-Cosine (NEGATIVE)
How is the total energy against time represented on a graph in an ideal simple harmonic system?
A straight line through the y axis
Ep+Ek = constant
How do you derive an equation for kinetic energy equation using SHM equations?
Ek = 1/2.m.v²
V = -ω.A.Sin(ωt)
1/2.m.ω².A².Sin²(ωt)
How do you derive an equation for maximum kinetic energy equation using SHM equations?
Ek = 1/2.m.v²
Vmax = ω.A
1/2.m.ω².A²
On an energy against time graph, describe the curves of Ep and Ek.
Ek = Inverse parabola
Ep = Parabola (x²)
Why can a mass spring system undergo SHM? Can you derive it?
The acceleration is directly proportional to the negative displacement (opposite direction).
F=ma F=-kx (k is the spring constant)
a=-k/m . x
Where in the cycle is the particle’s kinetic energy a maximum?
Centre/ origin of oscillations
Where in the cycle is a particle’s potential energy a maximum?
Maximum displacement of the oscillation
What are the limitations of a vertical spring system?
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.
What type of spring system can the time period equation T =2π √(m/k) ?
Horizontal with no friction
For a simple pendulum, how do you calculate the angular velocity?
ω=√(g/l)
For a simple pendulum, how do you calculate the time period?
T=2π√(l/g)
Why does a simple pendulum undergo SHM?
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
For a horizontal spring system with multiple springs, how do you calculate the Time period and Angular velocity?
Original equations but add the total spring constants of the multiple springs.
T=2π√(m/k1+k2)
ω=√(k1+k2/m)
For a horizontal spring system how do you calculate the Time period and Angular velocity?
T=2π√(m/k)
ω=√(k/m)
What is damping?
The reduction in energy and amplitude of oscillations due to resistive forces on the oscillating system.
What happens too the time period during light dampening?
Nothing, it remains constant
What happens to the amplitude of oscillations during light dampening?
Decays exponentially with time
Describe the forces acting on an oscillator during light dampening
Resistive forces oppose the motion, Restorative forces bring the oscillator back to equilibrium
What is critical damping?
The oscillator returns to rest in the shortest time possible, without oscillating.
What is heavy damping?
The oscillator return to equilibrium without any oscillations over a longer period of time (over damping).
What is the driving frequency?
Frequency of forced oscillations (By a driving force)
What is the natural frequency?
The frequency of oscillation when the system is allowed to oscillate freely.
When does the resonance peak occur?
Natural frequency = Driving frequency
What happens at the resonance peak?
Amplitude increases significantly due to the most efficient energy transfer between Ek and Ep
What effect does dampening have on the natural frequency of a system?
None!
What effect does dampening have on the Amplitude of resonance?
Reduces the Amplitude
What effect does dampening have on the resonant frequency of a SHSystem ?
The resonant frequency becomes < Natural frequency
On an A/f graph, shit too the left and less steep peak
As the degree of damping increases, what happens to the resonance peak?
The peak of the curve lowers and it becomes broader.Furthermore, the peak of resonance itself moves further to the left.
