6.1 Further Mechanics

Question 1

define linear velocity

Answer 1

rate of change of linear displacement

Question 2

define angular velocity

Answer 2

rate of change of angular displacement

Question 3

what are the two standard equations for linear velocity and angular velocity

Answer 3

linear velocity : change in displacement / time
angular velocity : change in theta / time

Question 4

what is the linear velocity of the ball that rolls 5.0 cm every 0.25 seconds

Answer 4

use equations w = v / r
and w = 2 x Pi x f
remember f = 1 / t
make them equal to each other
v = (2 x pi x r) / x t
so v = 0.126 ms ^-1

Question 5

convert 500 000 rpm to rad s^-1
b) calculate angular velocity when turbine rotates at 500 000 rpm

Answer 5

500 000 rpm / 60 = 8333 rev s^-
8333 x 2 Pi = 52360 rad s^-1
b) f = 500 000 / 60 = 8333 Hz
w = 2Pi x 8333 = 52360 rad s^-1

Question 6

what must act on a ball to cause it to move in a circular path

Answer 6

an acceleration is always caused by a resultant force so a resultant force must be acting on it

Question 7

What would happen to the ball if the string snapped ?

Answer 7

• it will move along a straight line tangential path to where it was cut with constant velocity
• from the side it will move in a parabolic arc as if in freefall

Question 8

for a ball in a string, which force contributes to the centripetal force

Answer 8

tension in the string

Question 9

for a car on a flat road surface, which forces contribute to the centripetal force

Answer 9

friction between tyres and the road surface

Question 10

Describe what happens if the car enters the bend at a higher speed
b) If the road surface is wet, describe the adjustments should the driver make

Answer 10

required centripetal force is greater than max friction
b) max friction will decrease
so max linear velocity will decrease
driver will need to enter bend slower

Question 11

what are the two conditions for simple harmonic motion

Answer 11

1. Acceleration acts in the opposite direction to displacement. The acceleration of the object is always directed towards the equilibrium position.
2. The acceleration is always proportional to the displacement of the object from the equilibrium position

Question 12

define free oscillation

Answer 12

a free oscillation is one in which there are no external forces

Question 13

when do damped oscillations occur

Answer 13

when frictional or viscous forces (resistive forces produced by fluids) act on object

Question 14

how does damped oscillations differ from free oscillations

Answer 14

• time period unaffected
• each amplitude decreases with each oscillation because system loses energy working against frictional forces

Question 15

what are damping forces proportional to

Answer 15

velocity squared
- the force acts in the opposite direction to velocity

Question 16

what is a lightly damped system

Answer 16

the resisting force is small and energy is transferred to the surroundings very slowly. The system oscillates and the amplitude of oscillation reduces only gradually.

Question 17

what is a critically damped system

Answer 17

the energy is transferred to the surroundings very rapidly. The oscillator does not actually oscillate at all before coming to rest. Examples of this are in car suspensions or moving coil analogue meters where you don’t want oscillations to occur.

Question 18

what is a heavily damped system

Answer 18

the damping force is very large and the system does not oscillate, but only slowly returns to the equilibrium position. The system has almost no kinetic energy, only potential energy.

Question 19

define resonance

Answer 19

• driver frequency is equal to natural frequency
• large / maximum amplitudes of vibrations of mass
• as there is a maximum trsansfer of energy from support rod
• driver force displacement has 90 degrees phase difference

Question 20

what is natural frequency

Answer 20

the frequency the object will oscillate if there are no external forces acting on it

Question 21

what is driving frequency

Answer 21

the frequency when a system is made to oscillate by a periodic external

Question 22

what happens when
a) driving force is less than natural force
b) they are equal
c) natural force is less than driving force

Answer 22

a) low amplitude oscillations
similar amplitude to driving force
in phase with driving force
b) resonance occurs
large amplitude oscillations
much bigger than driving force
90 degrees out of phase
c) low amplitude oscillations
180 degrees out of phase

Question 23

Two pendulums A and B oscillate with simple harmonic motion.
The time period of A is 2.00 s and the time period of B is 1.98 s.
A and B are released in phase.
What is the number of oscillations of A before A and B are next in phase?

Answer 23

1.98 / 2-1.98 = 1.98 / 0.02 = 99 (for A)
b would be 100

Question 24

A body performs simple harmonic motion.
What is the phase difference between the variation of displacement with time and the variation of
acceleration with time for the body?

Answer 24

Pi radians / 180 degrees
acceleration is proportional to - displacement
(same magnitude opposite directions)

Question 25

Suggest why the chevron separation on motorways does not take into account the distance travelled as a car comes to rest after the brakes are applied.

Answer 25

It is assumed that the car in front would take the same time/travel the same distance as
the car behind when braking/ only difference is reaction time of the driver of car behind

Question 26

The loudspeaker creates variations in pressure and produces a sound wave in the air around it.
State the type of wave produced and describe the motion of the particles in this type of wave.

Answer 26

longitudinal as they oscillate along direction of energy transfer

Question 27

State the conditions for simple harmonic motion.

Answer 27

The acceleration is proportional to the displacement ✔
the acceleration is in opposite direction to displacement✔

Question 28

A small quantity of fine sand is placed onto the surface of the plate. Initially the sand grains stay in contact with the plate as it vibrates. The amplitude of the vibrating surface remains
constant at 3.5 × 10−4 m over the full frequency range of the signal generator. Above a particular frequency the sand grains lose contact with the surface.
Explain how and why this happens.

Answer 28

• when the vibrating surface accelerates down with an acceleration less than the acceleration of free fall the sand stays in contact
• above a particular frequency, the acceleration is greater than g
• there is no contact force on the sand OR
  sand no longer in contact when downwards acceleration of plate is greater than acceleration of sand due to gravity

Question 29

Discuss the motion of the ball in terms of the forces that act on it. In your answer
you should:
* explain how Newton’s three laws of motion apply to its motion in a circle
* explain why, in practice, the string will not be horizontal.
You may wish to draw a diagram to clarify your answer.
The quality of your written communication will be assessed in your answer

Answer 29

First law: ball does not travel in a straight line, so a force must be
acting on it
* although the ball has a constant speed its velocity is not constant
because its direction changes constantly
* because its velocity is changing it is accelerating
* Second law: the force on the ball causes the ball to accelerate (or
changes the momentum of it) in the direction of the force
* the acceleration (or change in momentum) is in the same direction
as the force
* the force is centripetal: it acts towards the centre of the circle
* Third law: the ball must pull on the central point of support with a
force that is equal and opposite to the force pulling on the ball from
the centre
* the force acting on the point of support acts outwards
* Support of ball: the ball is supported because the rope is not
horizontal
* there is equilibrium (or no resultant force) in the vertical direction
* the weight of the ball, mg, is supported by the vertical component
of the tension, F cos θ, where θ is the angle between the rope and
the vertical and F is the tension
* the horizontal component of the tension, F sin θ, provides the
centripetal force m ω^2r

Question 30

a rider rides a rotor where they are in a cylinder where it keeps being rotated
Explain why the riders slide down the wall as the ride slows down.

Answer 30

centripetal force decreases
as friction decreases

Question 31

At the maximum speed the centripetal acceleration is 29 m s–2
when the diameter = 4.5m
Show that the maximum angular velocity of a rider is 3.6 rad s–1

Answer 31

a = w x A
Amplitude = radius
29 = w^2 x r
w = 3.59

Question 32

what is meant by forced vibrations

Answer 32

vibrations are forced when periodic force is applied frequency determined by frequency of driving force
amplitude is small at a high frequency and large at a low frequency
frequency of driver and displacement is out of phase by 180 degrees

Question 33

Which one of the following statements best describes how the shape of the curve would
differ if the damping had been greater?
A the curve would be lower at all frequencies
B the curve would be higher at all frequencies
C the curve would be unchanged except at frequencies above the resonant frequency
where it would be lower
D the curve would be unchanged except at frequencies above the resonant frequency
where it would be higher

Answer 33

A

Question 34

Which one of the following statements always applies to a damping force acting on a
vibrating system?
A It is in the same direction as the acceleration.
B It is in the opposite direction to the velocity.
C It is in the same direction as the displacement.
D It is proportional to the displacement.

Answer 34

B

Question 35

To celebrate the Millennium in the year 2000, a footbridge was constructed across the
River Thames in London. After the bridge was opened to the public it was discovered that
the structure could easily be set into oscillation when large numbers of pedestrians were
walking across it.
a) What name is given to this kind of physical phenomenon, when caused by a
periodic driving force?
b) Under what condition would this phenomenon become particularly hazardous?
Explain your answer

Answer 35

a) forced vibrations or resonance
b) driving force is at same frequency as natural frequency so resonance occurs
- large amplitude vibrations produced or large energy transfer to structure
- could cause damage to structure or bridge to fail

Question 36

Suggest two measures which engineers might adopt in order to reduce the size of
the oscillations of a bridge

Answer 36

• stiffen the structure by reinforcement
• install dampers or shock absorbers

Question 37

Explain what is meant by damping.

Answer 37

damping when force opposes motion or damping removes energy

Question 38

What effect does damping have on resonance?

Answer 38

damping reduces sharpness of resonance
or reduces amplitude at resonant frequency

Question 39

For a body performing simple harmonic motion, which one of the following statements is correct?
A The maximum kinetic energy is directly proportional to the frequency.
B The time for one oscillation is directly proportional to the frequency.
C The speed at any instant is directly proportional to the displacement.
D The maximum acceleration is directly proportional to the amplitude.

Answer 39

D

Question 40

Answer 40

D

Question 41

Answer 41

D

Question 42

A particle of mass 5.0 × 10^–3 kg, moving with simple harmonic motion of amplitude 0.15 m, takes 47 s to make 50 oscillations.
What is the maximum kinetic energy of the particle?

Answer 42

frequency = number of oscillations / time taken
f = 50 / 47 = 1.064 Hz
w = 2 x Pi x f = 6.686 rad/s
Vmax = w x A
Ke = 1/2 x mass x w^2 x A^2
Ke = 2.5 x 10^-3

Question 43

A simple pendulum has a time period of 1.42 s on Earth. The gravitational field strength at the
surface of Mars is 0.37 times that at the surface of the Earth.
What is the time period of the pendulum on Mars?

Answer 43

Use of equation T = 2Pi x root length / gravitational field strength
1.42 x 1 / root 0.37 = 2.33 seconds

Question 44

A satellite X is in a circular orbit of radius r about the centre of a spherical planet of mass M. What is the speed in terms of G,M and r

Answer 44

F = GMm / r^2
F = mv^2 / r
re arrange to get v = Root GM / r

Question 45

An example of an equation to find total energy using a model in simple harmonic motion

Answer 45

E = 1/2 m^2 w^2 A^2
As Vmax = wA

Question 46

Answer 46

A

Question 47

A mass on the end of a spring undergoes vertical simple harmonic motion. At which point(s)
is the magnitude of the resultant force on the mass a minimum?
A at the centre of the oscillation
B only at the top of the oscillation
C only at the bottom of the oscillation
D at both the top and bottom of the oscillation

Answer 47

A

Question 48

A baby bouncer consisting of a harness and elastic ropes is suspended from a doorway. When
a baby of mass 10 kg is placed in the harness, the ropes stretch by 0.25 m. When the baby
bounces, she starts to move with vertical simple harmonic motion. What is the time period of her
motion?

Answer 48

Use of equations
T = 2pi x Root m/k
& F = k x change in length & F = mg
to find k, mg = kL
k = 392
T = Pi x root 10/392
= 1.004 seconds

Question 49

Which one of the following statements is true when an object performs simple harmonic
motion about a central point O?
A The acceleration is always directed away from O.
B The acceleration and velocity are always in opposite directions.
C The acceleration and the displacement from O are always in the same direction.
D The graph of acceleration against displacement is a straight line.

Answer 49

D

Question 50

Answer 50

A

Question 51

A mechanical system is oscillating at resonance with a constant amplitude. Which one of the following statements is not correct?
A The applied force prevents the amplitude from becoming too large.
B The frequency of the applied force is the same as the natural frequency of oscillation of the
system.
C The total energy of the system is constant.
D The amplitude of oscillations depends on the amount of damping.

Answer 51

A

Question 52

For a particle moving in a circle with uniform speed, which one of the following statements is
correct?
A The kinetic energy of the particle is constant.
B The force on the particle is in the same direction as the direction of motion of the particle.
C The momentum of the particle is constant.
D The displacement of the particle is in the direction of the force.

Answer 52

A

Question 53

A particle travels at a constant speed around a circle of radius r with centripetal acceleration
a. What is the time taken for ten complete rotations?

Answer 53

T = Circumference of circle / speed
= 2Pi / v
Total time = 10T
equation for centripetal acceleration = v^2 / r
v = root ar
T = 2pi / root ar
Time taken for 10 complete oscillations = 20Pi x root ar

Question 54

Which one of the following statements always applies to a damping force acting on a
vibrating system?
A It is in the same direction as the acceleration.
B It is in the same direction as the displacement.
C It is in the opposite direction to the velocity.
D It is proportional to the displacement.

Answer 54

C

Question 55

a disc of diameter 120mm that can turn about an axis through its centre.
The disc is turned through an angle of 30° in 20 ms. What is the average speed of a point on the
edge of the disc during this time?

Answer 55

arc length equation
s = radius x theta
convert 30 degrees to radians = Pi / 6 radians
s = Pi / 6 x 0.06 = 0.01 Pi
v average = s / t
0.01 Pi / 20 x 10^-3
= 0.5 Pi m/s

Question 56

A mass on the end of a string is whirled round in a horizontal circle at increasing speed until
the string breaks. The subsequent path taken by the mass is
A a straight line along a radius of the circle.
B a horizontal circle.
C a parabola in a horizontal plane.
D a parabola in a vertical plane.

Answer 56

D

Question 57

A body moves with simple harmonic motion of amplitude 0.90 m and period 8.9 s. What is the
speed of the body when its displacement is 0.70 m?

Answer 57

Use equation
v = w times by root A^2 - x^2
v = 0.40

Question 58

A mass M hangs in equilibrium on a spring. M is made to oscillate about the equilibrium
position by pulling it down and releasing it. The time for M to travel back to the equilibrium
position for the first time is 0.50 s.
What is the time period

Answer 58

0.5 seconds = t/4
because
This cycle has four equal parts, corresponding to distinct points in the oscillation:

First quarter: From the maximum displacement
(+A) to the equilibrium position.
Second quarter: From the equilibrium position to the opposite maximum displacement (−A).
Third quarter: From
−A back to the equilibrium position.
Fourth quarter: From the equilibrium position back to +A.

hence time period =

Question 59

Which one of the following statements concerning forced vibrations and resonance is
correct?
A An oscillating body that is not resonating will return to its natural frequency when the
forcing vibration is removed.
B At resonance, the displacement of the oscillating body is 180° out of phase with the forcing
vibration.
C A pendulum with a dense bob is more heavily damped than one with a less dense bob of the
same size.
D Resonance can only occur in mechanical systems.

Answer 59

A

Question 60

Answer 60

A

Question 61

.For a particle moving in a circle with uniform speed, which one of the following statements is
incorrect?
A The velocity of the particle is constant.
B The force on the particle is always perpendicular to the velocity of the particle.
C There is no displacement of the particle in the direction of the force.
D The kinetic energy of the particle is constant.

Answer 61

A

Question 62

3.A body undergoes forced oscillation. Which one of the following will not be increasedby increasing
the amplitude of the oscillatory driving force?
A the amplitude of the driven oscillation
B the energy of the driven oscillation
C the frequency of the driven oscillation
D the power required to maintain the driven oscillation

Answer 62

c

Question 63

What is the area under acceleration - time graph

Answer 63

Change in velocity

Question 64

A particle is oscillating with simple harmonic motion described by the equation:
s = 5 sin (20πt)
How long does it take the particle to travel from its position of maximum displacement to its
mean position

Answer 64

max displacement, A = 5
mean position, s = 0
T = 2π / w
= 2π / 20π
T = 0.1
t = T / 4
= 0.025 seconds

Question 65

Answer 65

C

Question 66

Describe the energy changes which take place during one complete oscillation of a vertical mass-spring system, starting when the mass is at its lowest point.

Answer 66

elastic pe → kinetic energy → gravitational pe / elastic

Question 67

Answer 67

0.8 seconds
During one oscillation there are two energy transfer cycles

Question 68

Answer 68

0.8 seconds
During one oscillation there are two energy transfer cycles

Question 69

A simple pendulum of time period 1.90 s is set up alongside another pendulum of time period 2.00 s. The pendulums are displaced in the same direction and released at the same time.
Calculate the time interval until they next move in phase. Explain how you arrive at your answer.

Answer 69

(n+1) x 1.9 = n x 2
n = 19 oscillations of longer pendulum
there is 20 oscillations on shorter pendulum
minimum time between phase condition = 38 seconds (19 x 2)

Question 70

Answer 70

Question 71

Answer 71

Question 72

Answer 72

Question 73

Answer 73

D
At d = 0, t=0
Ek max and velocity max

Question 74

Answer 74

Question 75

Answer 75

a max less than 1
Curve more spread out
D

Question 76

Answer 76

Question 77

Suggest an advisory speed for this section of the motorway

Answer 77

Question 78

Answer 78

Question 79

Part C of question on next flashcard

Answer 79

Question 80

Part C of question from previous flashcard

Answer 80

Question 81

Answer 81

Question 82

Answer 82

Question 83

Answer 83

Question 84

Answer 84

Question 85

Answer 85

Question 86

Answer 86

Question 87

Answer 87

Question 88

Part b) explain your answer

Answer 88

Question 89

When the length of a simple pendulum is decreased by 600mm, the period of oscillation is halved. What is the original length of the pendulum

Answer 89

Question 90

Answer 90