Further Mechanics and Thermal Physics Flashcards
What is angular displacement?
If a wheel takes T seconds to rotate once, it will turn through an angle of 2π/T rad each seconds.
The frequency of each rotation f = 1/T.
Therefore, the angular displacement:
θ = 2πt/T or θ = 2πft
What is angular velocity?
w = θ/t = 2π/T = 2πf
for one complete rotation:
time = distance / velocity = 2πr/v = 2π/w
therefore, w = v/r
What is centripetal acceleration?
A bike wheel rotates with constant speed but is constantly changing direction.
W = Fd but the F is at right angles to the velocity therefore no work is done and no energy is transferred.
However, the velocity is constantly changing direction, therefore the bike is always accelerating.
This is called centripetal acceleration.
a = v²/r = w²r
What is centripetal force?
Since the bike is accelerating, there must be force.
This is called the centripetal force.
F = mv²/r = mw²r
The centripetal force increases if …
F = Δp/Δt Δp increases i.e. -mass increases -speed increases Δt decreases i.e. -radius decreases
Vertical circle?
AT TOP F = mg + T but CF = mv²/r therefore, T = mv²/r - mg AT BOTTOM F = T - mg but CF = mv²/r therefore, T = mv²/r + mg
Cars going over bumps/hills/bridges?
The car is supported by a force S. When v = 0, S = mg. As v increases, the car has a greater tendency to continue in a straight line. This is because S decreases. F = mg - S but CF = mv²/r therefore, S = mg - mv²/r As v increases, a greater centripetal force is required to keep the car moving in a circle. This happens by S decreasing.
Cars going round bends?
The centripetal force is provided by the friction of the tyres on the road.
The car has a certain v(max) before it begins to slip on the road because the friction is not enough to provide the centripetal force.
friction(max) = mv(max)²/r
Conical circle/pendulum?
T(H) = Tsinθ T(V) = Tcosθ The forces are balanced vertically, therefore: T(V) = mg Tcosθ = mg The resultant force, F = T(H) = Tsinθ But this is the CF, therefore, Tsinθ = mv²/r therefore, tanθ = v²/rg
Banked tracks?
A car is travelling round a bend and the track is banked at an angle, θ.
If there’s no friction on the tyres, the CF is provided by the horizontal component of R.
F = Rsinθ (also equal to mv²/r)
mg is balanced by the vertical component of R so:
mg = Rcosθ
therefore, tanθ = v²/rg
What is the equilibrium position?
The lowest point in an oscillating motion.
What is displacement, x?
The distance from the equilibrium position.
What is amplitude, A?
The maximum displacement.
What is period, T?
The time taken to complete one full oscillation.
What is frequency, f?
The number of oscillations per unit time.
What is phase difference?
The fraction of an oscillation between the position of two oscillating objects.
Δt/T x 2π
What is angular frequency, w?
The rate of change of angular position.
2πf
What is an oscillating object?
An oscillating object moves repeatedly in one way, then in the opposite direction, through its equilibrium position.
The displacement of the object from its equilibrium position continually changes throughout motion.
What do the graphs of x, v and a look like?
x: cos graph
v: flipped sin graph
a: flipped cos graph
What is the gradient of the x-t graph?
The v-t graph.
The magnitude of v is greatest when the gradient of the x-t graph is greatest i.e. x = 0
The magnitude of v is 0 when the gradient of the x-t graph is 0 i.e. x = max
What is the gradient of the v-t graph?
The a-t graph.
The magnitude of a is greatest when the gradient of the v-t graph is greatest i.e. v = 0 and x = max
The magnitude of a is 0 when the gradient of the v-t graph is 0 i.e. v = max and x = 0
What is simple harmonic motion?
A type of oscillation where the acceleration of the oscillator is directly proportional to the displacement from the equilibrium position, in the opposite direction to the displacement.
a ∝ -x
An oscillator in SHM is an isochronous oscillation, so the period of the oscillation is independent of the amplitude.
What is the equation for acceleration in SHM?
The constant of proportionality depends on the time period, T of the oscillation.
The shorter the time period, the faster the oscillation, therefore the larger the acceleration at any given displacement. Therefore, the larger the constant.
Therefore, a = -w²x
a(max) = w²A
What is the equation for displacement in SHM?
x = Asinwt (if oscillator begins at equilibrium position) x = Acoswt (if oscillator begins at amplitude position)
What is the equation for velocity in SHM?
v = ±w√A² - x² v(max) = wA
What is the equation for a mass-spring system?
F = -kx if Hooke's law is obeyed but F = ma therefore ma = -kx a = (-k/m)x so SHM but a = -(2πf)²x therefore (2πf)² = k/m so f = √k/m / 2π but T = 1/f therefore T= 2π√m/k
What is the equation for a simple pendulum?
Resolving vertically: Tcosθ = mg Resolving horizontally: -Tsinθ = F -Tsinθ = ma therefore -tanθ = a/g but for small angles tanθ = sinθ = x/l therefore, -x/l = a/g hence a = -(g/l)x but a = -(2πf)²x therefore (2πf)² = g/l so f = √g/l / 2π but T = 1/f therefore T= 2π√l/g
What energy transfer occurs in SHM?
If there’s no friction, free oscillation occurs.
The energy is constantly changing between Ek and Ep.
But the total energy remains constant i.e. energy is always conserved.
The max Ek occurs at equilibrium, where v = max.
The max Ep occurs at the the amplitude position, where x = max.
RECALL GRAPH
What is damping?
The process by which the amplitude of the oscillations decreases over time. This is due to energy loss due to resistive forces such as drag or friction.
Successive waves become smaller.
However, the frequency always remains the same.
What is light damping?
The time period, T stays constant.
The amplitude decreases by the same fraction each cycle.
What is heavy damping?
The damping is so strong, that the object takes a long time to return to equilibrium.
Oscillation does not occur.
What is critical damping?
The object returns to equilibrium in the shortest time possible and does not overshoot.
What is an example of damping?
A car suspension system includes a spring and a damper. The damper provides almost critical damping so that oscillation dies away quickly after going over a bump.
What is a free vibration?
The total energy stays the same over time therefore, the amplitude of the vibration is constant.
No external forces are applied and the object is oscillating at its natural frequency.
This is a theoretical idea because in real systems, energy is dissipated to the surroundings and the amplitude decays to 0 (damping).
What is a forced vibration?
A periodic force is applied to an object, which causes it to oscillate at a particular frequency
What is resonance?
If applied frequency = natural frequency:
-The amplitude of the oscillations becomes very big.
-Phase difference between the displacement and periodic force changes to π/2.
NB: Below resonance, they are in phase and about resonance, the phase difference is π.
What is the effect of damping on resonance?
The resonant frequency is f0.
If there is no damping, the amplitude will continue to increase until the system fails.
As damping is increased, the amplitude will decrease at all frequencies, and the max amplitude will occur at a lower frequency.
Examples of resonance?
- A swing. Initially, the motion is slow and the swing doesn’t reach its maximum potential. Once it reaches its natural frequency of oscillation, a gentle push helps it maintain that amplitude of swing all throughout due to resonance.
- Barton’s Pendulums. The driver pendulum is displaced and released, and the other pendulums undergo forced oscillation as the effect of the oscillation is transported along the wire. The pendulum which is the same length as the driver has the same time period, and thus the same natural frequency. Therefore, this pendulum oscillates with a larger amplitude.
- Bridges. In the absence of damping, a crosswind or many people walking across the bridge, in step with each other apply a periodic force equal to the natural frequency of the bridge span. This can cause resonance, and the bridge can collapse.
