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4.2: Circular motion and oscillations
- One radian - is the angle subtended at the centre of a
circle by an arc of length equal to the circle's radius.
- The period T - of of an object in circular motion is the time taken to
complete one revolution. It's related to the speed v and radius r by
equation: v = 2πr/T
- v = 2πr/T
- v = 2πr/T
- A car going around a corner at constant speed is changing velocity because the
direction of the velocity is changing, although its magnitude remains the constant.
Vectors are needed to calculate numerical value for change in velocity.
- Centripetal acceleration - The CA, a, of an object travelling in a circle of
radius r, with constant speed v, is given by: a = v²/r
- CA - direction towards the centre of the circle
- a = v²/r
- CA - direction towards the centre of the circle
- F = ma
- F = mv²/r
- T = mv²/r
- Tension = Force
- Tension = Force
- T = mv²/r
- F = mv²/r
- Centripetal force - The resultant force on a object, acting towards the centre of the circle causing it
to move in a circular path. Given by F = mv²/r
- Examples of circular motion
- 1) ball on a string
- Smooth friction surface, circular path r and
the force provided is through the tension in
the string.
- T = mv²/r
- T = mv²/r
- Smooth friction surface, circular path r and
the force provided is through the tension in
the string.
- 2) A pendulum
- Same as
string with
no frictional
surface
- tanθ = v²/Rg
- g = a = 9.81ms
- g = a = 9.81ms
- tanθ = v²/Rg
- Same as
string with
no frictional
surface
- 3) Theme park ride
- When a person is at the top of the
ride, the weight is directed towards
the centre of the circle. So
accelertion = weight + contact
force.
- When at bottom, weight acts in the
opposite direction to acceleration.
Therefore due to the centripetal force
being of the same magnitude
anywhere on the ride, the contact
force of the seat must be greater. So
contact force = centripetal force +
weight.
- When at bottom, weight acts in the
opposite direction to acceleration.
Therefore due to the centripetal force
being of the same magnitude
anywhere on the ride, the contact
force of the seat must be greater. So
contact force = centripetal force +
weight.
- When a person is at the top of the
ride, the weight is directed towards
the centre of the circle. So
accelertion = weight + contact
force.
- 1) ball on a string
- Gravitational field strength
- Field - A field is the region in which a force operates
- Gravitational field strength - The force acting per unit mass at any point.
- g = F/m
- The relationship of GFS and accelaration at free fall due to
gravity are both equal to 9.81. However the difference is that
they have different unit. GFS = Nkg^-1 and free fall =
9.81ms^-1.
- Field - A field is the region in which a force operates
- Newtons law of gravitation
- The gravitational force of attraction between two
bodies is directly proportional to the product of
their masses and inversely proportional to the
square of the distance between them.
- F = -GMm/r²
- minus sign only used to
establish it is an
attractive force, unlike
magnetic that attract
and repel.
- Large r, larger
distance by
squared amount.
So triple radius
means 1/9th of
gravitational
force on a object.
- minus sign only used to
establish it is an
attractive force, unlike
magnetic that attract
and repel.
- F = -GMm/r²
- g = GM/r²
- Satellites
- Rotating around the
Earth due to its
distance from Earth.
- Kepler's third law
- T² is proportional to the r^3
- The period (T) squared of a
planet is proportional to the
mean radius (r) cubed.
- T² = (4π²r²/GM)
- T² is proportional to the r^3
- Geostationary orbit
- 1) Orbit centred on the centre of the Earth
- 2) Travel west to east
- 3) Equator
- 4) Period 24 hours
- 3) Equator
- 2) Travel west to east
- Telecommunications
- Costly and high power
(signals) needed to function
- High cost as lot of energy needed to
break free of 'gravity well'
- High cost as lot of energy needed to
break free of 'gravity well'
- Costly and high power
(signals) needed to function
- Low-level satellites
- Less expensive
- Greater detail photos as
lower radius (closer to earth)
- Higher intensity (power per unit area)
can be achieved on Earth's surface.
- Higher intensity (power per unit area)
can be achieved on Earth's surface.
- Greater detail photos as
lower radius (closer to earth)
- Less expensive
- Weather, spying, mapping,
global positioning.
- Movement of individual
people outdoors, effect
of deforestation,
shrinking ice caps, drying
of inland sea, urban
expansion can be seen.
- Movement of individual
people outdoors, effect
of deforestation,
shrinking ice caps, drying
of inland sea, urban
expansion can be seen.
- 1) Orbit centred on the centre of the Earth
- Rotating around the
Earth due to its
distance from Earth.
- Astronauts weightless
due to no support force,
accelerating towards the
Earth (still experience
gravity).
- Satellites
- The gravitational force of attraction between two
bodies is directly proportional to the product of
their masses and inversely proportional to the
square of the distance between them.
- Oscillations
- SHM - When the acceleration a of an object is
proportional to its displacement x and the acceleration
is in the opposite direction to the displacement.
- a ∝ -x
- a = -(2πf²) x
- x = Asin(2πft) OR x = Acos(2πft)
- x = Asin(2πft) OR x = Acos(2πft)
- a = -(2πf²) x
- a ∝ -x
- Free oscillations - Object oscillates without driving force acting.
Objects undergoing FO vibrate at their natural frequency.
- Graphical analysis
- Displacement/time = sine
curve... a/t = refelction in x...
v/t = gradient of x/t.
- a/x = negative
proportional
- v/x = Circle, A and v + or -
- v/x = Circle, A and v + or -
- a/x = negative
proportional
- Displacement/time = sine
curve... a/t = refelction in x...
v/t = gradient of x/t.
- Energy interchanges in SHM
for pendulum. KE and GPE
fluctuate. Total energy
constant.
- Damping
- Damping - Oscillation in which the
KE is converted into other forms
and so the amplitude of the
oscillations drop.
- Critical damping - D of oscillating
systems when forces cause the
system to return to the equilibrium
position without oscillating. (e.g.
pendulum through thick treacle).
- LightD and HeavyD (pendulum
through air/water respectively).
- LightD and HeavyD (pendulum
through air/water respectively).
- Critical damping - D of oscillating
systems when forces cause the
system to return to the equilibrium
position without oscillating. (e.g.
pendulum through thick treacle).
- Damping - Oscillation in which the
KE is converted into other forms
and so the amplitude of the
oscillations drop.
- Resonance
- Resonance - When the
driving frequency is equal
to the natural frequency of
an oscillating system. This
causes a dramatic increase
in the amplitiude of the
oscillations.
- Effect of damping on
resonance is that
increasing the amount
of damping reduces the
amplitude and
frequency of the driven
oscillator.
- Uses
- Construction industry, so buildings don't collapse.
- Car industry, eliminate rattling/bouncing.
- Radio/television, adjusting resonant frequency to equal signal.
- Nuclei of atoms (MRI colour scanning) resonate in a field of suitable magnetic oscillations.
- Nuclei of atoms (MRI colour scanning) resonate in a field of suitable magnetic oscillations.
- Aircraft industry,
wings don't bounce
so much when
landing or take off.
- Radio/television, adjusting resonant frequency to equal signal.
- Car industry, eliminate rattling/bouncing.
- Construction industry, so buildings don't collapse.
- Uses
- Effect of damping on
resonance is that
increasing the amount
of damping reduces the
amplitude and
frequency of the driven
oscillator.
- Resonance - When the
driving frequency is equal
to the natural frequency of
an oscillating system. This
causes a dramatic increase
in the amplitiude of the
oscillations.
- SHM - When the acceleration a of an object is
proportional to its displacement x and the acceleration
is in the opposite direction to the displacement.
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