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. 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