5 Circular motion

Question 1

radian

Answer 1

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

Question 2

radians equation

Answer 2

angle in radians= arc length/radius

Question 3

360 degrees in radians

Answer 3

for a complete circle, the arc length is equal to the circumference of the circle
angle in radians= 2pir/r = 2pi radians

Question 4

angular velocity

Answer 4

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

Question 5

centripetal force

Answer 5

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

Question 6

constant speed in a circle

Answer 6

v=d/t
v=2pir/T
since ω=2pi/T
v=rω

Question 7

centripetal acceleration

Answer 7

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

Question 8

centripetal force equation

Answer 8

F=ma and a=v^2/r
F=mv^2/r
since v=ωr
F=mω^2r

Question 9

investigating circular motion

Answer 9

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

Question 10

sources of centripetal force

Answer 10

friction
tension
gravitational attraction

Question 11

banked surfaces

Answer 11

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

Question 12

cyclist in velodrome free body diagram

Answer 12

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

Question 13

conical pendulum

Answer 13

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

Question 14

oscillating motion

Answer 14

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

Question 15

displacement

Answer 15

distance from the equilibrium position

Question 16

amplitude

Answer 16

the max displacement from the equilibrium position

Question 17

period

Answer 17

time taken to complete one full oscillation

Question 18

frequency

Answer 18

the number of complete oscillations per unit time

Question 19

angular frequency

Answer 19

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

Question 20

simple harmonic motion

Answer 20

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

Question 21

converting degrees into radians

Answer 21

multiply the angle by 2pi and divide by 360

Question 22

converting radians into degrees

Answer 22

multiply radian by 360 and divide by 2pi

Question 23

using graphs to demonstrate SHM

Answer 23

taking a simple pendulum as an example
displacement-time graphs
velocity-time graphs
acceleration-time graphs

Question 24

displacement-time graph to demonstrate SHM

Answer 24

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

Question 25

velocity-time graph to demonstrate SHM

Answer 25

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

Question 26

acceleration-time graph to demonstrate SHM

Answer 26

inverted version of displacement-time graph
max negative acceleration at max positive displacement.
zero displacement at zero displacement

Question 27

displacement(-time graph) equations

Answer 27

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

Question 28

velocity(-time graph) equations

Answer 28

v= +/- w√(A2 -x2)
vmax= wA

Question 29

graphs of energy and displacement

Answer 29

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

Question 30

damping

Answer 30

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

Question 31

light damping

Answer 31

small damping forces- causes the amplitude of the oscillator to gradually decrease with time, but the period of the oscillations is almost unchanged

Question 32

heavy damping

Answer 32

for large damping forces, the amplitude decreases significantly, and the period of the oscillations also increases slightly

Question 33

free oscillations

Answer 33

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

Question 34

forced oscillatons

Answer 34

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)

Question 35

what happens if the driving frequency is equal to the natural frequency

Answer 35

the object will resonate

causes the amplitude of the oscillations to increase dramatically

Question 36

resonance

Answer 36

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