PHYSICS A2 UNIT 4 SECTION 1: FURTHER MECHANICS

CONSERVATION OF MOMENTUM

CIRCULAR MOTION

SIMPLE HARMONIC MOTION

RESONANCE

# Question AS MECHANICS THEORY

1 What are vector/scalar quantities? • A vector has magnitude and direction • A scalar quantity has magnitude and direction 2 What are 4 examples of vector quantities? • Displacement, velocity, force and acceleration 3 How would you solve the parallel component of a force? • 𝐹𝑐𝑜𝑠𝜃 4 How would you solve the perpendicular component of a force? • 𝐹𝑠𝑖𝑛𝜃 5 What is velocity? • The rate of change of displacement 6 How would you find velocity from a graph? • The gradient of a displacement-time graph 7 What is acceleration? • The rate of change of velocity 8 How would you find acceleration from a graph? • The gradient of a velocity-time graph 9 What do the S, U, V, A, T symbols represent? • S= displacement, meters • U= initial velocity, ms-1 • V= final velocity, ms-1 • A= acceleration, ms-2 • T= time taken, seconds 10 What is the work done on an object? • The work done on an object is the force applied for a distance travelled 11 How would you find the work done on an object where θ is the angle between the

direction of the force and the displacement of an object? • The work done on a object is given by 𝐹𝑠𝑐𝑜𝑠𝜃, where 𝜃 is the angle between the direction of the

force and the displacement of the angle 12 What is the kinetic energy of an object? • !

!𝑚𝑣!

13 What is the gravitational potential energy of an object? • 𝑚𝑔∆ℎ 14 What is power? • The rate of transfer of energy

FORCES AND MOMENTUM 15 What is momentum? • The linear momentum of an object is defined as it’s mass multiplied by its velocity 16 What is the unit of momentum?

• The unit is kg ms-1 • Or Newton seconds Ns 17 Is momentum a scalar or vector quantity? • Momentum and velocity are both vectors, because they have magnitude and direction 18 What is Newton’s First Law? • An object remains at rest or in a uniform motion unless acted on by a force 18.1 How does Newton’s First Law relate to momentum? • Tells us that a force is needed to change the velocity, and therefore the momentum of an object • If the momentum of an object is constant, there is no resultant force acting on it • If the mass of an object is constant and the object has constant momentum, it follows that the

velocity of the object is also constant • If a moving object with constant momentum gains or loses mass, however, its velocity would

change to keep its momentum constant 19 What is Newton’s Second Law? • The rate of change of momentum of an object is proportional to the resultant force on it • In other words, the resultant force is proportional to the change of momentum per second 20 How does force link to momentum? • F=ma 21 How is Newton’s Second Law derived from SUVAT? • For an object of constant mass m acted on by a constant force F • It’s initial momentum = 𝑚𝑢 • Its final momentum = 𝑚𝑣 • It’s change of momentum = 𝑚𝑣 −𝑚𝑢 • According to Newton’s second Law, the force applied is proportional to the change of

momentum per second • Therefore 𝐹 = !!!"#$ !" !"!#$%&!

!"#$ !"#$%= !"!!"

!= !(!!!)

!

• 𝑎 = !!!!

• 𝐹 = 𝑚𝑎 • So 𝐹 = ∆!"

∆!

22 What happens to momentum when no external forces act on it? • Assuming no external forces act on it, linear momentum is always conserved • This means the total linear momentum of two objects before they collide equals the total linear

momentum after the collision 23 What is an impulse? • The impulse of a force is defined as the force multiplied buy the time for which the force acts 24 What is the equation for an impulse? • Therefore, the impulse = 𝐹∆𝑡 = ∆(𝑚𝑣) 25 What is the area under a Force vs. Time graph? • Newton’s second Law gives 𝐹 = !"!!"

!

• Rearranging this gives 𝐹𝑡 = 𝑚𝑣 −𝑚𝑢 • The area under the line represents the impulse of the force 𝐹𝑡 which is equal to 𝑚𝑣 −𝑚𝑢 • In other words, the area under the line of a force-time graph represents the change of

momentum or the impulse of the force 26 What is the force of impact? • The force of impact is the change of momentum on an object per the time for which its in contact

with the object 27 What is the equation for Force of Impact? • 𝐹 = !!!"#$ !" !"!#$%&!

!"#$%!$ !"#$= !"!!"

!

27.1 How can the average force of impact be worked out from a force-time graph? • The overall change in momentum (area under the graph) divided by the contact time

28 What happens to the impact force for an object in rebound? • When an object hits a wall, it rebounds • It’s momentum changes direction due to the impact • If the object hits the wall normally (at 90o), it rebounds normally • Hence the direction of the momentum is just reversed • The velocity and therefore the momentum after the impact is in the opposite direction to the

velocity before the impact • And it therefore has a negative sign • Initial momentum = 𝑚𝑢 • The final momentum has a reversed velocity, so is = −𝑚𝑣 • 𝑇ℎ𝑒 𝑖𝑚𝑝𝑎𝑐𝑡 𝑓𝑜𝑟𝑐𝑒 𝐹 = !!!"#$ !" !"!#$%&!

!"#$%!$ !"#$= !!" !(!")

!

29 What is Newton’s Third Law? • When two objects interact, they exert equal and opposite forces on each other • If object A exerts a force on object B, there must be an equal and opposite force action on

object A due to object B 30 What is the Principle of Conservation of Momentum? • The Principle of Conservation of Momentum states that for a system of interacting objects, the

total momentum remains constant, provided no external resultant force acts on the system 31 How can you test the Principal Conservation of Momentum? • Using a motion sensor linked to a computer • The mass of each trolley is measured before the test • With trolley B at rest, trolley A is given a push so it moves towards trolley B at constant velocity • The two trolleys stick together on impact • The computer records and displays the velocity of trolley A throughout this time • The computer display shows that the velocity of trolley A dropped suddenly when the impact

took place • The velocity of trolley A immediately before the collision, uA, and after the collision, V, can be

measured • The measurements should show that the total momentum of both trolleys after the collision is

equal to the momentum of trolley A before the collision • In other words 𝑚! +𝑚! 𝑉 = 𝑚!𝑢! +𝑚!𝑢! 32 What is an elastic collision? • An elastic collision is where there is no loss of kinetic energy 33 What is conserved in an elastic collision? • The kinetic energy just after the impact must equal its kinetic energy just before impact • There is therefore no loss of kinetic energy 34 What is an example of an elastic collision? • Very low speed impacts between two cars is almost perfectly elastic, provided no damage is

done 35 What is an inelastic collision?

● Totally ● Partially

• A totally inelastic collision is one where the colliding objects stick together • A partially inelastic collision is where the colliding objects move apart, and have less kinetic

energy after the collision than before 36 What is conserved in an inelastic collision? • Linear momentum 37 How is momentum affected in an explosion? • When two objects fly apart after being initially at rest, they recoil from each other with equal and

opposite amounts of momentum • So they move away from each other in opposite directions • Total initial momentum = 0

• Total momentum immediately after the explosion • Momentum of A + momentum of B • 𝑚!𝑣! +𝑚!𝑣! • 𝑚!𝑣! +𝑚!𝑣! = 0 • 𝑚!𝑣! = −𝑚!𝑣! • The minus sign means that the two masses move apart from each other in opposite directions

UNIFORM CIRCULAR MOTION 38 What is Uniform Circular Motion? • An object rotating at a steady rate is said to be in uniform circular motion 39 How do you find the speed of rotation? • 𝑠𝑝𝑒𝑒𝑑 = !"#$%&'(

!"#$

• The circumference of the circle is 2𝜋𝑟 • The frequency of rotation is 𝑓 = !

! where T is the time for one rotation

• Therefore, 𝑣 = !!"!

39.1 How do you convert between radians and degrees? • Angle in radians = angle in degrees x !!

!"#

40 What is the Angular displacement? • The angle an object in circular motion turns through 41 What equation finds Angular Displacement? • The angular displacement of an object in time t is therefore given by 𝜃 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = !!"

!= 2𝜋𝑓𝑡

• Where T is the time for one rotation • And t is the amount of time for which the object is rotating 42 What is the Angular Speed? • The rate of change of angular displacement of an object in circular (or orbital) motion • Also defined as the angular displacement per second 43 What equation finds the Angular Speed? • 𝜔 = !"#$%!& !"#$%&'()(*+

!"#$ !"#$%= !

!= !!

!= 2𝜋𝑓

44 What is the frequency of rotation? • Frequency is the number of complete revolutions per second – in Hertz • 𝑓 = !

!

45 What is the time period of rotation? • The time period, T, is the time taken for one complete revolution • 𝑇 = !

!

46 How would you calculate angular speed using frequency? • 𝜔 = 2𝜋𝑓 47 How would you calculate angular speed using time period? • 𝜔 = !!

!

48 How is an object in uniform circular motion accelerating? • The velocity of an object moving round a circle at constant speed continually changes direction • Because velocity changes, the object therefore accelerates 49 What is centripetal acceleration? • The velocity of an object in uniform circular motion at any point is along the tangent to the circle

at that point • The direction of the velocity changes continuously as the object moves round on its circular path • The change in the direction of the velocity is towards the centre of the circle • So its acceleration is towards the centre of the circle and is referred to as centripetal

acceleration 50 What is the equation for finding acceleration in circular motion?

• Centripetal acceleration, 𝑎 = !!

!

51 What is the proof for finding acceleration in circular motion? • Consider an object in uniform circular motion at speed 𝑣 moving in a short time interval 𝛿𝑡 from

position A to position B along the perimeter of the circle with radius 𝑟 • Therefore, the distance AB along the circle, 𝛿𝑠 = 𝑣𝛿𝑡 • The line from the object to the centre of the circle at C turns through an angle of 𝛿𝜃 when the

object moves from A to B • The velocity direction of the object turns through the same angle 𝛿𝜃 • The change of velocity, 𝛿𝑣 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑎𝑡 𝐵 − 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑎𝑡 𝐴

• The triangles ABC and the velocity vector triangle have the same shape because they both

have two sides of equal length with the same angle 𝛿𝜃 between the two sides

• Provided 𝛿𝜃 is small, then !"!= !"

!

• Because 𝛿𝑠 = 𝑣𝛿𝑡 then !"!= 𝑣 !"

!

• Therefore, acceleration 𝑎 = !!!"#$ !" !"#$%&'(!"#$ !"#$%

= 𝛿𝑣𝛿𝑡 =

𝑣2𝑟

52 What is the centripetal force? • To make an object move around in a circular path, it must be acted on by a resultant force which

changes its direction of motion • The resultant force on an object moving round a circle at constant speed is referred to as the

centripetal force because it acts in the same direction as the centripetal acceleration, which is towards the centre of the circle

• Any object that moves in circular motion is acted on by a resultant force which always acts towards the centre of the circle

• This resultant force is the centripetal force and therefore causes centripetal acceleration

53 What are the 2 equations for finding the centripetal force? • 𝐹 = 𝑚𝑎

• 𝑎 = 𝑣2𝑟 = 𝜔!𝑟 (where 𝜔 = !

!)

• Therefore, 𝐹 = 𝑚𝑣2𝑟 = 𝑚𝜔!𝑟

54 What is the support force for an object in circular motion? • 55 How would you find the support force for an object going over a hill? • At the top of the hill, the support force S from the road on the vehicle is directly upwards in the

opposite direction to its weight, mg • The resultant force on the vehicle is the difference between the weight and the support force • This difference acts towards the centre of curvature of the hill as the centripetal force

• 𝑚𝑔 − 𝑆 = 𝑚𝑣2𝑟

56 How would you find the force of friction for an object moving round a roundabout? • Consider a vehicle of mass m moving at a speed of v in a circle of radius r • It moves round a roundabout on a level road • The centripetal force is provided by the force of friction between the vehicle’s tyres and the road

surface

• 𝐹𝑜𝑟𝑐𝑒 𝑜𝑓 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛, 𝐹 = 𝑚𝑣2𝑟

• For no skidding or slipping to occur, the force of friction between the tyres and the road surface must be less than a limiting value of 𝐹!, which is proportional to the vehicles weight

56.1 Why may a vehicle skid on a bend is the speed is too high?

• 𝐹 = 𝑚𝑣2𝑟

• If speed is greater than a certain value, the centripetal force will be greater than the value of friction

• This will make the vehicle skid/slip in the direction of the velocity 57 Through which wheels does the centripetal force act on a train and why? • Centripetal force acts on the outer wheels • The flange of the outer wheel pushes outwards against the curved outer rail as the engine

attempts to carry on moving in a straight line. • The outer rail therefore pushes inwards on this same flange, providing the centripetal force 58 What provides the centripetal force on a vehicle travelling on a banked track? • Without any banking, the centripetal force on a road vehicle is provided only by sideways friction

on the tyres • A vehicle on a bend slips outwards if its speed is too high 59 Why can speed be higher on a banked track? • On a banked track, speed can be higher • For there to be no sideways friction on the tyres due to the road, the horizontal component of

the support forces N1 and N2 must act as the centripetal force

• Resolving these into horizontal components ( 𝑁! + 𝑁! 𝑠𝑖𝑛𝜃) and vertical components ( 𝑁! +

𝑁! 𝑐𝑜𝑠𝜃)

• 𝑁! + 𝑁! 𝑠𝑖𝑛𝜃 act towards the centre of the path, so act as the centripetal force

• Hence, 𝑁! + 𝑁! 𝑠𝑖𝑛𝜃 =𝑚𝑣2𝑟

• Because ( 𝑁! + 𝑁! 𝑐𝑜𝑠𝜃) balance the weight, 𝑁! + 𝑁! 𝑐𝑜𝑠𝜃 = 𝑚𝑔

• Therefore 𝑡𝑎𝑛𝜃 = !!!!! !"#$!!!!! !"#$

= 𝑚𝑣2𝑚𝑔𝑟

• Simplifying this gives 𝑡𝑎𝑛𝜃 = 𝑣2𝑟

• In other words, there is no sideways friction if the speed v is such that 𝑣! = 𝑔𝑟𝑡𝑎𝑛𝜃 60 What is the support force at the bottom of a rollercoaster?

• 𝑆 −𝑚𝑔 = 𝑚𝑣2𝑟

61 How does the kinetic energy vary with the gravitational potential on a very long swing?

• The maximum speed is when the swing passes through the lowest point (equilibrium position) • This can be worked out by equating the gain of kinetic energy to the loss of potential energy

• !!!

!= 𝑚𝑔ℎ

62 What is the speed at the lowest point of a very long swing? • As it passes through the lowest point, 𝑣! = 2𝑔ℎ 63 What is the support force at the bottom of the swing? • At the lowest point, the support force S on the person due to the swing is in the opposite

direction to the persons weight, mg

• Therefore 𝑆 −𝑚𝑔 = 𝑚𝑣2𝐿

• 𝑆 = 𝑚𝑣2𝐿 +𝑚𝑔

64 What is the support force at the top of the swing?

• 𝑚𝑔 + 𝑆 = 𝑚𝑣2𝐿

• 𝑆 = 𝑚𝑣2𝐿 −𝑚𝑔

65 Where is the support force (tension) greatest on a swing?

• At the bottom because 𝑆 = 𝑚𝑣2𝐿 +𝑚𝑔

SIMPLE HARMONIC MOTION 66 What is an oscillation? • Regular variation in magnitude or position about a central, equilibrium point • The displacement of the object from equilibrium continually changes during the motion 67 What is one full cycle of motion for an oscillation? • One full cycle is from maximum height at one side to maximum height at the other side and then

back to the original position again 68 Where is the equilibrium position found? • The equilibrium is the lowest point in the oscillation 69 What is the displacement of the object in an oscillation? • The distance from the equilibrium position 70 What is the amplitude of an oscillation? • The maximum displacement of the oscillating object from equilibrium 71 What are free oscillations? • If the amplitude is constant and there are no frictional (damping) forces present, the oscillations

are described as free oscillations 72 What is the time period for an oscillation? • T • The time for one complete cycle of oscillations • One full cycle after passing through any position, the object passes through the same position in

the same direction 73 What is the frequency of an oscillation? • The number of cycles per second • 𝑓 • Measured in Hertz 74 What is the angular frequency of an oscillation? • The angular frequency of an oscillation motion is defined as 𝜔 = !!

!

75 What is phase difference? • The phase difference is how much one oscillation lags behind another 76 How would you find the phase difference between two oscillations using their time

period? • = !!∆!

!

Where ∆𝑡 is the time between successive instants where the two objects are at maximum displacement in the same direction

77 How would you find the phase difference between two oscillations in radians? • !!∆!

!

78 What would the graph of displacement/time look like for an oscillation?

79 Describe the displacement-time graph for an oscillation. • At zero time, the object is suspended as far from its equilibrium as possible • This is the amplitude (maximum displacement) • When it is released, it falls back to the equilibrium position, hence there is no displacement • It then continues to swing to the maximum displacement (amplitude) on the other side of the

equilibrium point • This is shown as a negative displacement, as it travels in the other direction 80 What would the graph of velocity/time look like for an oscillation?

81 Describe the velocity/time graph for an oscillation. • At zero time, the object is held at its amplitude, hence there is no movement and no velocity of

the object • As it is released, it will gain speed in the negative direction, until is passes the equilibrium point,

where it has maximum velocity • As it moves away from the equilibrium, it will lose velocity, until it momentarily has zero at its

maximum amplitude again 82 What would the graph of acceleration/time look like for an oscillation?

83 Describe the acceleration/time graph for an oscillation. • Starts with maximum negative acceleration • At equilibrium point, it has zero acceleration • As the opposite equilibrium point, it has maximum positive acceleration • As it returns to the equilibrium, it is now decelerating 84 How do the graphs of displacement, velocity and acceleration against time link? • The differential of displacement graph is a velocity graph • The differential of a velocity graph is an acceleration graph • The integral of an acceleration graph is a velocity graph • The integral of a velocity graph is a displacement graph 84.1 When is the velocity greatest/zero? • The velocity is greatest when the gradient of the displacement-time graph is greatest (e.g. at

zero displacement when the object passes through the equilibrium) • The velocity is zero when the gradient of the displacement-time graph is zero (i.e. at maximum

displacement) 84.2 When is the acceleration greatest/zero? • The acceleration is greatest when the gradient of the velocity-time graph is greatest • This is when the velocity is zero and occurs at maximum displacement in the opposite direction • The acceleration is zero when the gradient of the velocity-time graph is zero • This is when the displacement is zero 85 How do acceleration and displacement link? • By comparing the displacement-time and the acceleration-time graphs, it can be seen that the

acceleration is always in the opposite direction to the displacement • In other words, if one direction is referred to as the positive direction and the other as the

negative direction, the acceleration direction is always the opposite sign to the displacement direction

86 How is Simple Harmonic Motion defined? • Simple Harmonic Motion is defined as oscillating motion in which the acceleration is:

1. Proportional to the displacement 2. Always in the opposite direction to the displacement

• Acceleration, a = -constant x displacement, x 87 What is the equation for Simple Harmonic Motion? • 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛,𝑎 = −(2𝜋𝑓)!𝑥 88 What is the equation for the displacement of an object in an oscillation?

• 𝑥 = 𝐴𝑐𝑜𝑠𝜃(2𝜋𝑓𝑡) 89 How is the equation for the displacement of an object in an oscillation derived?

• The diagram shows that the displacement, 𝑥, of the ball is just equal to the horizontal

component of the ball’s position, 𝑥 = 𝑟𝑐𝑜𝑠𝜃 • The radius of the circle, 𝑟, is the same as the amplitude of the ball’s simple harmonic motion (as

viewed from the side), 𝐴 • This is the maximum displacement 𝑥, of the ball • The angular speed 𝜔 is defined as 𝜃 𝑡 so, 𝜃 = 𝜔𝑡 • The angular speed is also defined as 𝜔 = 2𝜋𝑓 • Combining these equations gives you an expression for the displacement, 𝑥, of any object

undergoing SHM • 𝑥 = 𝑟𝑐𝑜𝑠𝜃 = 𝐴𝑐𝑜𝑠 𝜔𝑡 = Acos(2πft) 89.1 What is the equation for the acceleration of an object moving with SHM? • 𝑎 = −(2𝜋𝑓𝑡)!𝑥 89.2 How is the equation for the acceleration of an object moving with SHM derived? • For circular motion, 𝑎 = 𝜔!𝑟 • The angular speed of the ball, 𝜔, is given by 𝜔 = 2𝜋𝑓 • Putting these two equations together gives the acceleration of the ball as 𝑎 = (2𝜋𝑓)!𝑟 • The acceleration of an object in SHM is then the horizontal component of this • Hence, 𝑎 = −(2𝜋𝑓)!𝑟𝑐𝑜𝑠𝜃 89.3 When does the objects maximum acceleration occur? • When its at its maximum displacement • Hence 𝑎!"# = (2𝜋𝑓)!𝐴 89.4 What is the equation for the velocity of an object moving with SHM? • 𝑣 = ±2𝜋𝑓 𝐴! − 𝑥! 89.5 When is the maximum velocity of an object moving with SHM? • The maximum velocity is when the object is passing through the equilibrium, hence 𝑥 = 0 • 𝑣!"# = 2𝜋𝑓𝐴

SIMPLE HARMONIC OSCILLATORS 90 What energy is involved in a mass-spring system? • An object in SHM exchanges potential energy and kinetic energy as it oscillates • Elastic potential energy for masses on springs 91 What energy is involved in a pendulum? • Gravitational potential energy for pendulums 91.1 What is a restoring force?

• For any oscillating object, the resultant force on the object acts towards the equilibrium position • The resultant force is described as a restoring force, as it always acts towards the equilibrium 91.2 What happens when the restoring force proportional to the displacement? • Provided the restoring force is proportional to the displacement from equilibrium, the

acceleration is proportional to the displacement and always acts towards the equilibrium • Therefore the object oscillates with simple harmonic motion 92 Draw and describe how energy is transferred over time during an oscillation.

• As the object moves towards the equilibrium position, the restoring force does work on the

object and so transfers some Ep to Ek • When the object is moving away from the equilibrium, all that Ek is transferred back to Ep again • At the equilibrium position, the object’s Ep is zero and its Ek is maximum – therefore its velocity

is maximum • At the maximum displacement (the amplitude) on both sides of the equilibrium, the objects Ep is

maximum and its Ek is zero – so it’s velocity is zero 93 What is the energy of the object at its amplitude? • The gravitational potential energy is maximum • So the kinetic energy is zero 94 What is the sum of the potential and kinetic energy called? • The mechanical energy • It stays constant as long as the motion isn’t damped 94.1 How do you investigate the oscillations of a mass-spring system? • Use two stretched springs and a trolley • When the trolley is displaced then released, it oscillates backwards and forwards • The first half-cycle of the trolley’s motion can be recorded using a length ticker tape attached at

one end to the trolley • When the trolley is released, the tape is pulled through a ticker timer that prints dots on the tape

at a rate of 50 dots per second • This forms a displacement-time graph for the 95 How does adding extra mass effect the frequency of a mass-spring oscillation? • Adding extra mass to the trolley will reduce the frequency of the oscillations • Because extra mass will increase the inertia of the system • This is its resistance to change in speed • At a given displacement, the trolley would therefore be slower than if the extra mass had not

been added • Each cycle of oscillations will take longer 96 How does using a weaker spring affect the frequency of a mass-spring oscillation? • Also reduce the frequency of oscillations • The restoring force on the trolley at any given displacement would be less so the trolley’s

acceleration and speed at any given displacement would be less • Each cycle of oscillation will therefore take longer 97 What is Hooke’s Law?

How is the time period of oscillations found using this? • Hooke’s Law states that 𝐹 = −𝑘𝑥 • Where k is the spring constant and x is the displacement • Newton’s second Law states that the resultant force on an object equals the mass times the

acceleration • Acceleration, 𝑎 = −(2𝜋𝑓)!𝑥 • Combining these equations gives – 2𝜋𝑓 !𝑚𝑥 = −𝑘𝑥 • The x’s cancel to give 2𝜋𝑓 !𝑚 = 𝑘

• Rearranging this for frequency gives 𝑓 = !!!

𝑘 𝑚 • The reciprocal can then find the time period, T

• 𝑇 = 2𝜋 𝑚𝑘

98 How does a the frequency of a mass-spring system vary with spring stiffness?

• Using 𝑇 = 2𝜋 𝑚𝑘 ,

• If mass is constant, we can see that 𝑇 ∝ 1𝑘

• Therefore, 𝑇! ∝ 1 𝑘 • So, the frequency is increased if k is increased or if m is reduced 99 What is the equation for time period for a simple pendulum?

• 𝑇 = 2𝜋 𝑙 𝑔

100 How is the equation for time period for a simple pendulum derived?

• At small angles of 𝜃, the weight 𝑚𝑔 has vertical components of 𝑚𝑔𝑐𝑜𝑠𝜃 • And horizontal components of 𝑚𝑔𝑠𝑖𝑛𝜃 à this is towards the equilibrium position • The restoring force 𝐹 = −𝑚𝑔𝑠𝑖𝑛𝜃 • So the acceleration 𝑎 = !

!= !!"#$%&

!= −𝑔𝑠𝑖𝑛𝜃

• If 𝜃 doesn’t exceed 10o, aka it’s a small angle, then 𝑠𝑖𝑛𝜃 = !!= !

!

• Therefore, 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛,𝑎 = !!!𝑠 = −(2𝜋𝑓)!𝑠 (where (2𝜋𝑓)! = !

! )

101 What are free oscillations? • A freely oscillating object oscillates with a constant amplitude because there is no friction acting

on it • The only forces acting on it combine to provide the restoring force • They involve no transfer of energy to or from the surroundings 102 How does potential energy change with displacement for a pendulum system? • The potential energy curve is a parabolic shape • The kinetic energy curve is an inverted parabola 103 Therefore, what is the total energy of the pendulum system? • The sum of the kinetic energy and the potential energy is always equal to !

!𝑘𝐴! which is the

potential energy at maximum displacement • This is the same as the kinetic energy at zero displacement • So the two curves add together to give a flat line for the total energy 104 Why do pendulums eventually stop oscillating? • The oscillations of a simple pendulum gradually die away because air resistance gradually

reduces the total energy of the system (the mechanical energy) • In any oscillating system where friction or air resistance is present, the amplitude decreases 105 What are the forces responsible known as? • The forces causing the amplitude to decrease are known as dissipative forces because they

dissipate the energy of the system to the surrounding as thermal energy 106 What is damping? • The motion is said to be damped if dissipative forces are present 107 What is light damping? • Lightly damped systems take a long time to stop oscillating, and their amplitude only reduces

small amounts each period • The time period is independent of the amplitude, so each cycle takes the same length of time as

the oscillations die away • The amplitude of each oscillation gradually decreases, reducing by the same fraction each cycle

108 What is critical damping? • Critical damping reduces the amplitude in the shortest time possible • It is just enough to stop the system oscillating after it has been displaced and released from

equilibrium • The oscillating object returns to equilibrium in the shortest time possible without overshooting if

the damping is critical • Such damping is important in mass-spring systems such as vehicle suspension

109 What is heavy damping? • Heavy damping is where the damping is so strong that the displaced object returns to

equilibrium much more slowly than if the system is critically damped • No oscillation motion occurs • For example, a mass on a spring in thick oil would return to equilibrium very slowly after being

displaced and released

110 What is a periodic force on an oscillation? • Force applied to a free oscillations at regular intervals • E.g. when you time pushing someone on a swing so that the swing goes higher and higher 111 What is the natural frequency of an oscillation? • When a system oscillates without a period forced being applied to it, its frequency is referred to

as its natural frequency 112 What happens when a periodic force is applied to an oscillation? • When a periodic force is applied to an oscillating system, the response depends on the

frequency of the periodic force • The system undergoes forced oscillations when a periodic force is applied to it 113 What is the new frequency called after a period force has been applied? • The frequency of an oscillator with an applied force is called the applied frequency 114 What is resonance? • When the applied/driving frequency is equal to the natural frequency 115 What happens to the phase difference and amplitude as the applied/driving frequency

is increased from zero? • The applied frequency approaches the natural frequency of the mass-spring system • The amplitude of oscillations of the object increase more and more • The phase difference increases from zero to !

! at the natural frequency

116 What happens to the phase difference and the amplitude, as the applied/driving force equals the natural frequency of the mass-spring system?

• The amplitude of the oscillations become very large

• The lighter the damping on the system, the larger the amplitude becomes • The system in in resonance when the applied frequency equals the natural frequency • The phase difference between the displacement and the periodic force is !

! at resonance

• The periodic force is then exactly in phase with the velocity of the oscillating object 117 What happens as the applied force becomes greater than the natural frequency? • The amplitude of the oscillations decrease more and more • The phase difference between the displacement and the periodic force increases from !

! until the

displacement is 𝜋 radians out of phase with the periodic force 117.1 When is the amplitude of oscillations greatest? • When the applied frequency is equal to the natural frequency, providing the damping is light • Applied frequency of period force = the natural frequency of the system 118 How do Barton’s pendulums work?

• The diagram shows 6 pendulums hanging from a supporting thread which is stretched between

fixed points • A single driver pendulum (x) is the same length as one of the other pendulums also tied on the

thread (D) • The driver pendulum X is displaced and released so it oscillates in a plane perpendicular to the

plane of the pendulums at rest • The effect of the oscillating motion of D is transmitted along the support thread, subjecting each

of the other pendulums to forced oscillations • Pendulum D responds much more than any other pendulum • This is because it has the same length and therefore same time period/frequency as X • So its natural frequency is the same as the natural frequency of X • Therefore, D oscillates in resonance with X because it is subjected to forced oscillations of the

same frequency as its own natural frequency of oscillations • The response of each of the other pendulums depends on how close its length is to the length of

X 119 How may resonance occur on bridges? • A crosswind can cause a period force on a bridge span because of eddy currents created by the

wind along the bridge span • If the wind speed is such that the periodic force is equal to the natural frequency of the bridge

span, then resonance can occur in the absence of damping • Alternatively, a steady trail of people in step with each other walking across a footbridge can

cause resonant oscillations of the bridge span if there is insufficient damping • Soldiers marching in columns are taught to break step when crossing a footbridge to avoid

resonance 120 What is the amplitude / natural frequency graph for increasing degrees of damping?