5 Circular motion Flashcards
radian
SI unit for angle is the radian
a radian is the angle subtended by a circular arc with a length equal to the radius of the circle
this is an angle of approx 57.3 degrees for any circle
radians equation
angle in radians= arc length/radius
360 degrees in radians
for a complete circle, the arc length is equal to the circumference of the circle
angle in radians= 2pir/r = 2pi radians
angular velocity
the rate of change of angle for an object moving in a circular path
ω=θ/t
in a time t equal to one period T, the object will move through an angle theta equal to 2pi radians
ω=2pi/T
angular velocity measures in radians per second
as f=1/T
ω=2pif
centripetal force
any force that keeps a body moving with a uniform speed along a circular path is called a centripetal force
means centre seeking
a centripetal force is always perpendicular to the velocity of the object. so this force has no component in the direction of motion and so no work is done on the object. so its speed remains constant
constant speed in a circle
v=d/t
v=2pir/T
since ω=2pi/T
v=rω
centripetal acceleration
the acceleration of any object travelling in a circular path at constant speed
always acts towards the centre of the circle
a=v^2/r
v=rω so
a=ωr^2
centripetal force equation
F=ma and a=v^2/r
F=mv^2/r
since v=ωr
F=mω^2r
investigating circular motion
as a bung is swung in a horizontal circle the suspended weight remains stationary as long as the force it provides (mg) is equal to the centripetal force requires to make the bung travel in the circular path
if the centripetal force requires is greater than the weight then the weight moves upwards
a paperclip acts as a marker to make this movement clearer
the weight and thus the centripetal force required for different masses, radii, and speeds can then be investigated
sources of centripetal force
friction
tension
gravitational attraction
banked surfaces
the greater the speed of an object following a circular path, the greater the centripetal force required to make it follow this path
a car approaching a bend must slow down in order to ensure the max frictional force between the tyres and road is sufficient to provide the required centripetal force
if the car travels too fast it will follow a path of greater radius and leave the road
for the same reason the tracks in velodromes are banked up to 45 degrees so that track cyclists can travel at higher speeds
cyclist in velodrome free body diagram
weight, mg, acting down
normal contact force, N, perpendicular to floor
horizontal component of NCF, NH, acts towards centre of the circle
NH= N sinθ
conical pendulum
a simple pendulum that rotates at a constant speed, describing a horizontal circle
the time taken to complete each rotation depends only on the length of the pendulum string and the gfs
the horizontal component of the tension, FT, in the string N provides the centripetal force F required for the circular motion of the pendulum
F=ma=mv2/r
FT sinθ= mv2/r
the vertical component of the tension in the string must be equal to the weight of the pendulum bob as there is no acceleration in the vertical direction
FTcosθ = mg
sinθ/cosθ = tanθ tanθ = v2/rg
oscillating motion
repetitive motion of an object around its equilibrium point
consider an object that is displaced from its equilibrium position and then released. it travels towards its equilibrium position at increasing speed
it then slows down once it has gone past the equilibrium position and eventually reaches max displacement (amplitude).
it then returns towards its equilibrium position, speeding up, and once more slows down to a stop when it reaches max negative displacement
this motion is repeated over and over again
displacement
distance from the equilibrium position
amplitude
the max displacement from the equilibrium position
period
time taken to complete one full oscillation
frequency
the number of complete oscillations per unit time
angular frequency
term used to describe the motion of an oscillating object and is closely related to angular velocity of an object in circular motion
ω= 2pi/T or ω=2pif
simple harmonic motion
a common kind of oscillating motion defined as oscillating motion for which the acceleration of the object is directly proportional to its displacement and is directed towards some fixed point
a= -ω^2 x
where ω^2 is a constant for the object
converting degrees into radians
multiply the angle by 2pi and divide by 360
converting radians into degrees
multiply radian by 360 and divide by 2pi
using graphs to demonstrate SHM
taking a simple pendulum as an example
displacement-time graphs
velocity-time graphs
acceleration-time graphs
displacement-time graph to demonstrate SHM
at zero displacement the pendulum is at, or moving through, its equilibrium position
at the maximum displacements it is at the top of its swing
the pendulum is at its maximum positive displacement at time t=0
gradient is equal to the velocity of the oscillator
at max displacements the velocity is zero because the gradient of the graph is zero
the velocity and the gradient of the graph are at their max as the pendulum moves through its equilibrium position
velocity-time graph to demonstrate SHM
velocity is zero when displacement is max
max velocity when gradient of displacement-time graph is max, when pendulum is moving through its equilibrium
gradient is equal to acceleration
acceleration-time graph to demonstrate SHM
inverted version of displacement-time graph
max negative acceleration at max positive displacement.
zero displacement at zero displacement
displacement(-time graph) equations
x=Acoswt
x=Asinwt
-if object begins oscillating from its amplitude (e.g. a pendulum lifted from its equilibrium point and released), then at t=0 the object is at its positive amplitude, so use the cosine version
-if object begins oscillating from its equilibrium position (e.g. pendulum flicked from its equilibrium position), then at time t=0 the object is at its equilibrium position and its displacement is 0. in this case use the sine version
velocity(-time graph) equations
v= +/- w√(A2 -x2) vmax= wA
graphs of energy and displacement
for any object moving in SHM the total energy remains constant, as long as there are no losses due to frictional forces
at the amplitude the pendulum is briefly stationary and has zero kinetic energy
all its energy is in the form of potential energy (gravitational and elastic)
as the pendulum falls it loses potential energy and gains KE
it has a max velocity and so max KE as it moves through its equilibrium position
as the pendulum passes through the equilibrium position it has no potential energy
a graph of energy against displacement shows how the total energy of an oscillating system remains unchanged.
there is continuous interchange between PE and KE but the sum at each displacement is always constant and equal to the total energy
damping
an oscillation is damped when an external force that acts on it has the effect of reducing the amplitude of its oscillations
e.g. a pendulum moving through air experiences air resistance, which damps the oscillations until eventually the pendulum comes to rest
light damping
small damping forces- causes the amplitude of the oscillator to gradually decrease with time, but the period of the oscillations is almost unchanged
heavy damping
for large damping forces, the amplitude decreases significantly, and the period of the oscillations also increases slightly
free oscillations
when a mechanical system is displaced from its equilibrium position and then allowed to oscillate without any external forces
frequency is known as the natural frequency of the oscillator
forced oscillatons
one in which a periodic driver force is applied to an oscillator
in this case the object will vibrate at the frequency of the driving force (driving frequency)
what happens if the driving frequency is equal to the natural frequency
the object will resonate
causes the amplitude of the oscillations to increase dramatically
resonance
occurs when the driving frequency of a force oscillation is equal to the natural frequency of the oscillating object
the amplitude of the oscillation increases considerably
if the system is not damped, the amplitude will increase to the point at which the object fails
