circular motion and simple harmonic motion

Question 1

define radian

Answer 1

the angle subtended at the centre of a circle when the arc length is equal to the radius`

Question 2

define time period

Answer 2

the time taken for an object to complete 1 full circular orbit

Question 3

define frequency

Answer 3

the number of circular orbit completed per second

Question 4

stratergies to improve the accuracy of time period measurements

Answer 4

allow the object to settle by allowing it to complete a few circles
time 10 full circular orbits and then divided by 10
complete 3 measurements of time period and calculate an average
place a fiducial marker to mark the start and end of circular orbits
use a motion sensor with a data logger to measure many swings and analyse how period changes over time

Question 5

define and calculate angular velocity

Answer 5

the number of radians swept by the radius of an object in circular motion per second

w = 2π/T = 2πf

Question 6

describe the condition of uniform circular motion

Answer 6

when an object travels at constant speed with the resultant force always directed towards the centre

Question 7

convert between angular velocity and tangential velocity

Answer 7

v = 2πr / T = wr

Question 8

define and calculate centripetal acceleration

Answer 8

when the acceleration is always directed towards the centre of the circular orbit

a = v^2 / r = w^2r

Question 9

calculate centripetal force

Answer 9

(assuming objects mass is constant)
F = ma = mv^2 / r = mw^2r

Question 10

practically demonstrate the relationship between radius and time period

Answer 10

select a radius of orbit by measuring out a length of string with a fixed mass hung on the other end to create a centripetal tension force and whirl it around your head in uniform circular motion

let the mass settle for 2 full orbits and start timing when the mass passes by a fiducial marker. Measyre the time for 10 complete oscillations and divide by 10 to get the time period. Repeat this process twice for this length and then repeat the whole process for 6 more lengths

Plot a graph of T^2 against r which should be a straight line passing through the origin as the mass and tension force are constant

F = mw^2r = (m4π^2 / T^2) x r
=> T^2 = (m4π^2 / F) x r

Question 11

define displacement

Answer 11

how far the object is from the equilibrium position measured in metres (for a pendulum angular displacement can be given in radians)

Question 12

define amplitude

Answer 12

the maximum displacement of the object from the equilibirum position measured in metres

Question 13

define time period

Answer 13

the time taken to complete 1 full oscillation

Question 14

define frequency

Answer 14

the number of complete oscillations per second

Question 15

define angular velocity

Answer 15

the number of radians per second travelled by the object

Question 16

define phase difference

Answer 16

the difference between 2 points on a wave function expressed as an angle

Question 17

calculate angular frequency

Answer 17

w = 2π / T = 2πf

Question 18

define simple harmonic motion

Answer 18

when the magnitude of the acceleration is directly proportional to the displacement but in the opposite directione

Question 19

strategies to improve the accuracy of time-period measurements

Answer 19

allow the oscillations to settle before starting timing
time 10 full oscillations and then divided by 10
complete 3 measurements of time period and calculate average
place a fiducial marker to mark the start and end of oscillations at the equilibrium position
use a slow motion video camera

Question 20

show that an object will be in simple harmonic motion if its displacement can be expressed as x = Acos(wt) or x = Asin(wt)

Answer 20

a = -w^2x
=> derivative of x / t = -w^2x

there are 2 functions where their 2nd derivative is -w^2 times the original function which are: cosine and sine
=> if x = Acos(wt), a = -Aw^2cos(wt) = -w^2x
=> if x = Asin(wt), a = -Aw^2sin(wt) = -w^2x

Question 21

describe the energy changes starting from maximum displacement for a pendulum

Answer 21

the pendulum starts with maximum GPE which is transferred to kinetic energy and the equilibrium position and then back t GPE. This whole process repeats as it swings back towards the starting position, finishing with GPE.

Question 22

describe the energy change starting for maximum velocity for a mass-spring system

Answer 22

the mass starts with maximum kinetic energy which is transferred to elastic potential energy at maximum displacement which is then transferred back to kinetic energy as the mass comes back to the equilibrium position. This process then repeats in the opposite direction finishing back with kinetic energy at the equilibrium position

Question 23

define free vibration

Answer 23

when an oscillating sytem experiences no external force so mechanical energy is constant

Question 24

define damped vibration

Answer 24

when an oscillating system experiences an external force acting to decrease the mechanical energy

Question 25

define forced vibration

Answer 25

when a system has a periodic external force applied to it

Question 26

desrcribe the effects of damping on an oscillating sytem

Answer 26

the damping force will act in the opposite direction to the velocity, decreasing the mechanical energy and therefore decreasing the amplitude and maximum velocity

Question 27

define resonance

Answer 27

when a system is subjected to a periodic external force with a frequency equal to the systems natural frequency

Question 28

define natural frequency

Answer 28

the frequency at which a system would oscillate if given an initial displacement and then allowed to oscillate freely

Question 29

describe the observation from a batons pendulum with 1 pendulum providing a periodic driving force and 3 other connected pendula that are shorter, the same length and longer length than the driving pendulum

Answer 29

pendulum y has the same length as pendulum p and therefore has the same natural frequency as p. therefore y will be in resonance, have much larger amplitude and will oscillate with its displacement 90 degrees out of phase with the driving pendulum displacement.

pendulum x has a shorter length than p, therefore it has a larger natural frequency than P and so will be forced to oscillate below its natural frequency with a small amplitude and its displacement will be in phase with pendulum p.

pendulum z has a longer length than P, therefore it has a smaller natural frequency than P and so will be forced to oscillate above its natural frequency with a small amplitude and its displacement will be in anti phase with pendulum p.,

Question 30

How can a graph of a over be correct

Answer 30

Straight line through origin showing a µ x