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Module G484.2
Circular Motion and
Oscillations
ANSWERS
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6
7
5
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Lesson 11 questions – Centripetal Force
/27)………%……..
(
1.
By describing the meanings of the terms Centripetal Force and
Centripetal Acceleration explain how an object travels around a circular
path.
……… All objects that follow a curved path must have force acting towards
the centre of that curve.(1) We call this force the centripetal force.(1) All
objects that feel an unbalanced force will accelerate.(1) If that force is towards
the centre of a curve it is called the centripetal acceleration. (1)
…………………………………………………………………………………… (3)
2.
In each case below, state what provides the centripetal force on the object:
a)
a car travelling at a high speed round a sharp corner;
……… Friction between the tyres and the road. [1]
…………………………………………………………………………………… (1)
b)
a planet orbiting the Sun;
……… Gravitational force acting on the planet due to the Sun. [1]
…………………………………………………………………………………… (1)
c)
an electron orbiting the positive nucleus of an atom;
………… Electrical force acting on the electron due to the positive nucleus. [1]
…………………………………………………………………………………… (1)
d)
clothes spinning round in the drum of a washing machine.
……… The (inward) contact force between the clothes and the rotating drum. [1] (1)
3.
The object rotates at 15 revolutions per minute. Calculate the angular speed in
radian per second.
angular speed 
15 rev min –1  2
 1.6 rad s –1
–1
60 s min
(2)
4.
This question is about a rotating restaurant.
A high tower has a rotating restaurant that moves slowly round in a circle while
the diners are eating. The restaurant is designed to give a full 360° view of the
skyline in the two hours normally taken by diners.
a)
Calculate the angular speed in radians per second.
Angular speed 
angle in radians
2 radians

 8.7 10  4 rad s 1 .
time in seconds (2  60  60) seconds
(2)
b)
The diners are sitting at 20 m from the central axis of the tower.
Calculate their speed in metres per second.
Instantaneous velocity = 2πr/T
= 2πx20/(2x60x60)
0.017m/s (2)
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c)
Do you think they will be aware of their movement relative to the
outside? Explain your answer.
They may just be able to perceive it but it is unlikely – they would see the skyline
move at less than 2 cm each second.
…………………………………………………………………………………… (2)
5.
An aeroplane is circling in the sky at a speed of 150ms–1. The aeroplane
describes a circle of radius 20 km. For a passenger of mass 80kg inside
this aeroplane, calculate her centripetal acceleration;
a=v2 /r [1]
a=1502 /20000 [1]
a=1.125ms–2 ≈ 1.1ms–2 [1]
6.
a = ……1.1……………. ms-2 (3)
The diagram shows a stone tied to the end of a length of string. It is
whirled round in a horizontal circle of radius 80 cm.
The stone has a mass of 90 g and it completes 10 revolutions in a time of 8.2 s.
a) Calculate:
i)
the time taken for one revolution;
8.2/10 = 0.82
ii)
T = …0.82…………s (1)
the distance travelled by the stone during one revolution;
Distance = circumference of circle
distance = 2πr=2π×0.80 = 5.03m≈ 5.0m [1]
distance = ……5.0……….. m (1)
iii)
the speed of the stone as it travels in the circle;
speed, v = 5.03/0.82 [1];
v = 6.13ms–1 ≈ 6.1ms–1 [1]
v = ………6.1………….. ms-1 (2)
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iv)
the centripetal acceleration of the stone;
2
a=v /r (1)
a = 6.13 2/0.80 (1)
a = 47 (1)
v = ………47………….. ms-2 (3)
b)
What provides the centripetal force on the stone?
………Tension in the string………………………………………
…………………………………………………………………………………… (1)
c)
What is the angle between the acceleration of the stone and its
velocity?
……………90 degrees………………………………………………
…………………………………………………………………………………… (1)
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Lesson 12 questions – Centripetal Force
(
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1. Calculate the centripetal force in the following cases:
a)
a ball of mass 150 g is spun in a horizontal circle of radius 3m at 5 ms-1
F
= mv2/r (1)
= 0.15x25/3 (1)
b)
F
Force = …1.25 (1)……… N (3)
the Earth (mass 6x10 kg) orbits the Sun once every year (3x107 s),
orbit radius 1.5x1011 m
= [6x1024x(2xx1.5x1011/3x107)2]/1.5x1011 (1)
= [6x1024x(3.14x104)2]/1.5x1011 = 3.95x1022 N (1)
c)
F
24
-31
Force = …3.95x1022 (1)……… N (3)
kg) orbits a nucleus in 1.6x10-16s, orbit
an electron (mass 9x10
radius 10-10 m
= [9x10-31x(2xx10-10/1.6x10-16)2]/10-10 (1)
= [9x10-31x(3.93x106)2]/10-10 (1)
= 1.39x10-7 N (1)
Force = …1.39x10-7……… N (3)
2
The diagram shows a car of mass 850kg travelling on a level road in a
clockwise direction at a steady speed of 20ms–1 round a bend with radius of curvature
32m.
v
a
a)
On the diagram, draw an arrow to show the velocity of the car (label
this v) and another arrow to show the acceleration of the car (label this a).
(2)
b)
Write an equation for the centripetal acceleration a of the car moving
on a level road at a speed v round a bend of radius of curvature r.
a=v2/r
(1)
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F
c)
Calculate the centripetal force acting on the car.
=
=
m v2/r (1)
(850 x 400)/32 (1)
Force = …… 1.1 × 104 (1)…………N (3)
d)
State what provides the centripetal force in (c).
………………………………………………………………………………………….
………… Friction between the tyres and the road…………………….
………………………………………………………………………………………….
……………………………………………………………………………………… (1)
3
A rubber toy of mass 40 g is placed close to the edge of a spinning turntable.
The toy travels in a circle of radius 12 cm. The toy takes 0.85 s to complete one
revolution. For this toy, calculate:
a)
its speed;
v = …… 0.89……….. ms-1 (2)
b)
the centripetal force acting on it.
F = ……0.26…………… N (3)
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Lesson 14 questions – Examples of circular motion
1
Read the short passage before answering the question below.
Figure 1.1 shows a section of a mass spectrometer. A beam of identical positivelycharged ions, all traveling at the same speed, enters an evacuated chamber through a
slit S. A uniform magnetic field directed vertically out of the plane of the diagram
causes the ions to move along a semicircular path SPT. The beam exits the chamber
through the slit at T
.
Fig 1.1
a)
i)
On Fig.1.1, draw an arrow to indicate the direction of the force
on the ion beam at P.
(1)
ii)
Name the rule you would use to verify that the ions are
positively charged.
…………Fleming’s
LHR…………………………………………………………………
…………………………………………………………………………………………
(1)
iii)
Explain why the ions follow a circular path in the chamber.
……Constant force on charge caused by magnetic field………………………
……perpendicular to path of ions/ towards center of
circle………………………………
…………………………………………………………………………………………
……
…………………………………………………………………………………………
……
…………………………………………………………………………………………
(2)
b)
Describe and explain the changes to the path of the ions for a beam of
ions of greater mass but the same speed and charge.
……Larger
semicircle……………………………………………………………………
……from
F=ma……………………………………………………………………………
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……so same force but bigger mass means less centripetal acceleration……………
…………………………………………………………………………………………
……
…………………………………………………………………………………………
……
…………………………………………………………………………………………
……
…………………………………………………………………………………………
(3)
c)
The speed of the singly charged ions is 3.0 x 105ms-1 in the magnetic
field of flux density 0.60T. The mass of each ion is 4.0 x 10-26 kg and
the force on each ion in the beam in the magnetic field is about 3 x 1014
N.
Calculate the radius of the semicircular path.
F=mv2/r
r=(4.0 x 10-26 x (3.0 x 105)2)/3 x 10-14
Radius = ……0.125………. m (3)
2
A compact disc (CD) player varies the rate of rotation of the disc in order to
keep the track from which the music is being reproduced moving at a constant linear
speed of 1.30ms-1. Calculate the rates of rotation of a 12.0cm disc when the music is
being read from
a)
The outer edge of the disc. Give your answers in both (i) rad s-1. and
(ii) Rev min-1
i)v=rω
ω=v/r
= 1.3/0.06
= 21.7 rad s-1
ii)(21.7/2π)60
i)
rate of rotation = ……21.7………………….. rad s-1 (2)
ii) rate of rotation = ……207………………….. Rev min-1 (2)
b)
A point 2.55 cm from the center of the disc. Give your answers in both
(i) rad s-1. and (ii) Rev min-1
i)v=rω
ω=v/r
= 1.3/0.0255
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ii)(51/2π)x60
i)
ii)
rate of rotation = ………51……………….. rad s-1 (2)
rate of rotation = …………487…………….. Rev min-1 (2)
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Lesson 15 questions – Gravitational Fields
1
(
/8)…………%………….
Draw the gravitational field lines for the following:
a)
The Earth from a distance
(2)
b)
The Earth up close
(2)
c)
A similar sized but much denser planet than Earth from a distance
(2)
d)
A similar sized but much denser planet than Earth from up close
(2)
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Lesson 16 questions – Newton’s Law of Gravitation
(
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Useful data
G = 6.67  10–11 N m2 kg–2
Earth’s mass = 5.97  1024 kg
Moon’s mass = 7.34  1022 kg
Sun’s mass is 2.0  1030 kg
Radius of the Moon = 1.64  106 m
Radius of the Earth = 6.37  106 m
Earth–Moon distance = 3.8  105 km
Earth–Sun distance = 1.5  108 km
MOST
1.
F
You may sometimes find it difficult to get up from the sofa after
watching a TV programme. Assuming the force of gravity acts between
the centre of your body and the centre of the sofa, estimate the
attraction between you and your sofa.
GMm (6.67  10 11 N kg 2 m 2 )  60 kg  100 kg

 1.6  10 6 N.
2
2
r
0.5 m 
(2)
2.
Calculate the size of the gravitational pull of a sphere of mass 10 kg on
a mass 2.0 kg when their centres are 200 mm apart.
What is the force of the 2.0 kg mass on the 10 kg mass?
F
8
GMm (6.67  10 11 N kg 2 m 2 )  10 kg  2.0 kg

 3.3  10 N.
2
2
r
0.200 m 
The pull on the 10 kg mass will be equal but opposite in direction.
(2)
3.
r 
At what distance apart would two equal masses of 150 kg need to be
placed for the force between them to be 2.0  10–5 N?
Gm 2
G
m
 150 kg
F
F
6.67  10 11 N kg 2 m 2
2.0  10 5 N
 0.27 m.
(2)
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4.
Calculate the gravitational pull of the Earth on each of the following
bodies:
the Moon;
F
GMm (6.67  10 11 N kg 2 m 2 )  (5.97  10 24 kg)  (7.34  10 22 kg)

 2.0  10 20 N.
2
8
r2
3.8  10 m


(2)
satellite A with mass 100 kg at a distance from the Earth’s centre 4.2 
107 m;
F 
GMm
r2

(6.67  10 11 N kg 2 m 2 )  (5.97  10 24 kg)  100 kg
4.2  10 m
7
2
 23 N.
(2)
and satellite B mass 80 kg at a distance from the Earth’s centre 8.0 
106 m.
F 
GMm
r2

(6.67  10 11 N kg 2 m 2 )  (5.97  10 24 kg)  80 kg
8.0  10 m
6
2
 5.0  10 2 N.
(2)
5.
Show that the unit for G, the universal gravitational constant, can be
expressed as m3 s–2 kg–1.
N kg 2 m 2  kg m s 2 kg 2 m 2  m 3 s 2 kg 1 .
(3)
6.
F 
Calculate the weight of an astronaut whose mass (including spacesuit)
is 72 kg on the Moon?
GMm
r
2

(6.67  10 11 N kg 2 m 2 )  72 kg  (7.34  10 22 kg)
1.64  10 m
6
2
 1.3  10 2 N.
(2)
What is the astronaut’s weight on Earth?
F
GMm (6.67  10 11 N kg 2 m 2 )  72 kg  (5.97  10 24 kg)

 7.1  10 2 N.
2
2
6
r
6.37  10 m
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
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(2)
Comment on the difference.
It’s about 6 times
(1)
7.
a)
Show that pull of the Sun on the Moon is about 2.2 times larger
than the pull of the Earth on the Moon.
Sun–Moon
F 
GMm
r2

(6.67  10 11 N kg 2 m 2 )  (2.0  10 30 kg)  (7.34  10 22 kg)
1.5  10
11
m

2
 4.4  10 20 N.
Earth–Moon
F
GMm (6.67  10 11 N kg 2 m 2 )  (5.97  10 24 kg)  (7.34  10 22 kg)

 2.0  10 20 N.
2
8
r2
3.8  10 m

ratio of attraction s 
4.4  10 20 N
2.0  10 20 N

 2.2
(4)
b)
Why then does the Moon orbit the Earth?
The Moon does of course orbit the Sun, as part of the Earth–Moon system. You can think of
the Moon’s orbit of the Earth as superimposed on its orbit of the Sun.
(2)
9.
The American space agency, NASA, plans to send a manned mission
to Mars later this century. Mars has a mass 6.42 x 1023 kg and a radius 3.38 x
106 m. G = 6.67 x 1011 N m2 kg-2 (a) The mass of a typical astronaut plus
spacesuit is 80 kg. What would be the gravitational force acting on such an
astronaut standing on the surface of Mars?
F = (Gmastrom x Mmars) / r2
F = (6.67 x 1011 N m2 kg-2) x 80 kg x (6.42 x 1023 kg) / (3.38 x 106 m) 2 = 300 N
(2)
(b) State whether an astronaut on Mars would feel lighter or heavier than on
Earth.
Would feel lighter.
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(1)
10
Sketch a graph showing how the Earth’s gravitational field varies with
distance.
(3)
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Lesson 18 questions – Gravitational Fields
1
A binary star is a pair of stars that move in circular orbits around their
common centre of mass. For stars of equal mass, they move is the same circular orbit,
shown by the dotted line in the diagram. In this question, consider the stars to be point
masses situated at their centres at opposite ands of a diameter of the orbit.
a)
each star.
i)
Draw on the diagram arrows to represent the force acting on
(2)
ii)
Explain why the stars must be diametrically opposite to travel
in the circular orbit.
……For circular motion there must be centripetal force
……This force is toward the centre of the circle – in this case the attraction due to
gravity and so must be along the diameter of the circle making the planets opposite
each other.…………………………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
……………………………………………………………………………………… (2)
b)
Newton’s law of gravitation applied to the situation in the diagram
may be expressed as
F = GM2/4R2
State what each of the symbols listed below represent
F ……force of attraction between Masses/stars……………………………
G ……gravitational constant…………………………………………
M ……mass of a star……………………………………………………
R ……radius of orbit…………………………………………………… (2)(1 for 2 2
for 4)
c)
i)
Show that the orbital period T of each star is related to its speed
v by v=2πR/T.
v=distance/time = 2πR/T
(1)
ii)
Show that the magnitude of the centripetal force required to
keep each star moving in its circular path is
F = 4π 2MR/T2
F = mv 2/r
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v=2πR/T
F = m(2πR/T)2/r
F = 4π 2MR/T2
iii)
(2)
Use equations from (b) and (ii) above to show that the mass of
each star is given by
M = 16π 2R3/GT2
F = GM2/4R2
F = 4π 2MR/T2
2
4π MR/T2= GM2/4R2
M = 16π 2R3/GT2
(2)
d)
Binary stars separated by a distance of 1x1011m have been observed
with an orbital period of 100 days. Calculate the mass of each star.
1 day = 86400s
M = 16π 2R3/GT2
M = (16π 2(0.5x1011)3)/((6.6710-11)86400002)
Mass = ……4.0 x 1030……………..kg (2)
2
This question is about gravitational fields. You may assume that all the mass
of the Earth, or the Moon, can be considered as a point mass at its centre.
a)
It is possible to find the mass of a planet by measuring the gravitational
field strength at the surface of the planet and knowing its radius.
i)
Define gravitational field strength, g.
………Gravitational force per unit mass…………………………………..
…………………………………………………………………………………………..
……………………………………………………………………………………… (1)
ii)
Write down an expression for g at the surface of a planet in
terms of its mass M and its radius R.
g=GM/R2
(1)
iv)
24
Show that the mass of the Earth is 6.0x10 kg.
Radius of Earth = 6400km.
M=g R2/G
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M=9.81x 64002/6.6710-11
M=6.0x1024kg
(1)
b)
i)
Use the data below to show the value of g at the Moon’s
surface is about 1.7 Nkg-1.
Mass of Earth = 81 x mass of Moon
Radius of Earth = 3.7 x radius of Moon
2
g=GM/R
g=6.6710-11x(6.0x1024/81)/(6400000/3.7) 2
g=1.65 Nkg-1.
(2)
ii)
Explain why a high jumper who can clear a 2m bar on Earth
should be able to clear a 7m bar on the Moon. Assume that the high jump on the
Moon takes place inside a “space bubble” where Earth’s atmospheric conditions exist.
………ΔgE/ ΔgM = ΔhE/ ΔhM =5.8……………………………………………………..
………If high-jumoers centre of mass is at 1m
So high jumper lifts centre of mass 1m on earth to reach 2m…………………..
………On the Moon raises it 5.8m to make 6.8m
…………………………………………………………………………………………..
……………………………………………………………………………………… (3)
c)
The distance between the centres of the Earth and the Moon is 3.8 x
108 m. Assume that the moon moves in a circular orbit about the centre of the Earth.
Estimate the period of this orbit to the nearest day.
Mass of Earth = 6.0x1024kg
1 day = 86400s
F=mv2/R; F = m(2πR/T)2/R= GMm/ R2
(2πR/T)2/R= GM/ R2
T2=4π2R3/GM
22
M is mass of moon = 7.4 x 10
R = 3.8 x 108 m
Period = …………27…………… day (5)
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3
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Lesson 21 questions – Oscillations
(
/23)………..%………..
ALL
1
Estimate the time periods of each of the following motions and hence calculate
(to 1 s.f.) their frequencies.
a)
a child on a playground swing
period = ……1……….s (1)
frequency = ……1……… Hz (1)
b)
a baby rocked in its mother’s arms
c)
period = ……0.5……….s (1)
frequency = ……2.0……… Hz (1)
the free swing of your leg from your hip
period = ……0.67……….s (1)
frequency = ……1.5……… Hz (1)
2
The pendulum bob of a grandfather clock swings through an arc of length
196mm from end to end. The period of the swing is 2.00 s.
a)
Explain what is meant by the period of the swing.
…………The time
for 1 full oscillation……………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
……………………………………………………………………………………… (2)
b)
What is the amplitude of the swing?
Amplitude = …0.098………… m (1)
c)
What is the frequency of the bob?
Frequency = …0.5………… Unit ……Hz…… (2)
3
What is (i) the frequency (ii) the period of:
a)
the rise and fall of the sea
i)……0.000022……………. Unit …Hz……. (2)
ii)………45…………. Unit …ks…… (2)
b)
the beat of a heart
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c)
i)………1.17…………. Unit …Hz……. (2)
ii)……0.85……………. Unit …s…… (2)
piano strings which oscillate when middle C is played
i)………256…………. Unit …Hz……. (2)
ii)……3.9……………. Unit …ms…… (2)
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Lesson 23 questions – Defining SHM
(
/17)……….%.........
ALL
1
Write an explanation which you could give to a non scientist of what simple
harmonic motion is. Use a situation with which they will be familiar to illustrate your
explanation. 1 mark will be given for written communication.
………Explain what an oscillation
is………………………………………………………
………Force/acceleration proportional to negative
displacement…………………………
………Suitable example like a swing
………………………………………………………………………………………
…………………………………………………………………………………………
……
…………………………………………………………………………………………
……
…………………………………………………………………………………………
……
…………………………………………………………………………………………
……
…………………………………………………………………………………………
……
…………………………………………………………………………………………
……
……………………………………………………………………………………… (4
+1)
MOST
2
The equation defining linear s.h.m. is
a = - (constant)x
a)
What units must the constant have?
…………s2
………………………………………………………………………………
…………………………………………………………………………………………
……
…………………………………………………………………………………………
(1)
b)
Two s.h.m.s, A and B, are similar except that the constant in A is nine
times the constant B. Describe how these s.h.m.s differ.
……frequency of A = 3 times the frequency of B…………………………
……since constant = (2πf)2……………………………………………………………
…………………………………………………………………………………………
……
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…………………………………………………………………………………………
(2)
3
A body oscillates with s.h.m. described by the equation
x = (1.6m)cos(3πs-1)t
a)
What are:
i)
the amplitude?
…………A =
1.6m………………………………………………………………………
…………………………………………………………………………………………
(2)
ii)
the period of motion?
………3π =
2π/T…………………………………………………………………………
………So T=2/3s……………………………………………………………… (2)
b)
For t=1.5s, calculate
(i) the displacement of the body
displacement = ……0…………..m (1)
(ii) the velocity
velocity = ………15……….. ms-1 (2)
(iii) the acceleration of the body
acceleration = ………0……….. ms-2 (2)
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Lesson 24 questions – Graphical Analysis of SHM
(
/15)……….%.........
1
ALL
A motion in which the acceleration/force is
proportional to the displacement; directed towards
the centre of oscillation/equilibrium position
MOST
T = 0.25 s or f = 1/T; f = 4 (Hz)
-2
a = -4π2f2A ; = 4 x 9.87 x 16 x 0.005 ; = 3.2 (m s ) ecf a(ii)
3
The diagram shows three sinusoidal (sine-shaped) graphs for displacement x,
velocity v and acceleration a for a simple harmonic oscillator whose Amplitude is
5nm and frequency is 300 MHz.
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a)
Explain the relationship between:
i)
the x-t and v-t graphs
…………π/2 out of phase – velocity
leads………………………………………………
…………………………………………………………………………………………
…………velocity is the gradient of displacement………………………………
……………………………………………………………………………………… (2)
ii)
the v-t and the a-t graphs
………… π/2 out of phase – acceleration leads…………………………………
…………………………………………………………………………………………
………… acceleration is the gradient of velocity……………………………
………………………………………………………………………………………(2)
iii)
the x-t and the a-t graphs
…………… π out of phase ………………………………………………
…………………………………………………………………………………………
…………………………………………………………………………………………
……………………………………………………………………………………… (2)
b)
Show that the maximum speed of this oscillator is 9ms-1.
vmax = (2πf)A (1)
= 2πx300x106x5x10-9 (1)
= 9.42 ≈ 9ms-1 (1)
(3)
c)
Illustrate your answer to (iii) by sketching a graph of x against a.
+2r
-r
a
x
+r
(2)
-2r
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Lesson 25 questions – SHM
(
1
/20)….…..%.......
This question is about a mass-spring system.
Fig1.1 shows a mass attached to two springs. The mass moves along a horizontal tube
with one spring stretched and the other compressed. An arrow marked on the mass
indicates its position on a scale. Fig 1.1 shows the situation when mass is displaced
through a distance x from its equilibrium position. The mass is experiencing an
acceleration a in the direction shown. Fig 1.2 shows a graph of the magnitude of the
acceleration against the displacement x.
fig 1.1
fig1.2
ALL
a)
i)
State one feature from each fig 1.1 and fig 1.2 which shows that
the mass performs harmonic motion when released.
…1.1…… acceleration in opposite direction to displacement
…1.2……acceleration proportional to displacement
…………………………………………………………………………………………
……
…………………………………………………………………………………………
(2)
MOST
ii)
Use data from fig 1.2 to show that the frequency of the simple harmonic
oscillations of the mass is about 5Hz.
a=-(2πf)2x
grad = -(2πf)2
1000 / 4π2= f2
25.33 = 5.03 ≈ 5Hz
(3)
b)
The mass-spring system of fig 1.1 can be used as a device to measure
acceleration, called an accelerometer. It is mounted on a rotating test rig, used to
simulate large g-forces for astronauts. Fig 1.3 shows the plan view of a long beam
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rotating about its axis A with the astronaut seated at end B, facing towards A. The
accelerometer is parallel to the beam and is fixed under the seat 10m from A.
fig 1.3
ALL
i)
When the astronaut is rotating at a constant speed, the arrow
marked on the mass has a constant deflection. Explain why.
…………At constant angular speed there is a constant centripetal acceleration toward
the centre of the circle.………………………………………………………………
…………………………………………………………………………………………
……
…………………………………………………………………………………………
……
…………………………………………………………………………………………
(2)
ii)
Calculate the speed v of rotation of the astronaut when the
deflection is 50mm.
at 50mm a= 50 ms-2
r=10m
a=v2/r
500= v2
v = ……22.4…………. ms-1 (2)
2
A bored student (in a biology lesson) holds one end of a flexible plastic ruler
against the laboratory bench and flicks the other end, setting the ruler into oscillation.
The end of the ruler moves a total distance of 8.0cm as in the diagram and makes 28
complete oscillations in 10s.
SOME
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a)
What are the amplitude x0 and frequency f of the motion of the end of the
ruler?
28 oscillations in 10 seconds
One osc.=10/28
1 oscillation = 0.357s
f=1/T
x0 = ………0.04…………….. m (1)
f = ………2.8…………….. Hz (1)
ALL
b)
Use x= x0cos2πft to produce a table of values of x and t for values:
t/s = 0, 0.04, 0.08, 0.12, 0.16, 0.20, 0.24, 0.28, 0.32, 0.36
t/s
x/m
0
0.04
0.08
0.12
0.16
0.2
0.24
0.28
0.32
0.36
0
0.03051
0.00653
-0.02054
-0.03787
-0.03722
-0.0189
0.00838
0.03169
0.03996
(4)
Draw a graph on the attached graph paper of x against t and use it to find the
maximum speed at the end of the ruler.
(4)
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displacement/m
q2
0.05
0.04
0.03
0.02
0.01
0
-0.01 0
-0.02
-0.03
-0.04
-0.05
0.1
0.2
0.3
0.4
gradient =
0.7 (+- 0.05)
ms-1
time/s
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Lesson 26 27 questions – Damping
(
/8)………..%...........
ALL
1
The faulty suspension system of a car is tests. The body of the stationary car is
pushed down and released. Fig 1.1 shows how the vertical displacement of the car
varies with time after it has been released.
fig 1.1
a)
i)
The graph shows light damping. Sketch on the same graph
what a critically damped system would look like.
(2)
ii)
Define simple harmonic motion.
………acceleration proportional to displacement
………directed towards a fixed point
…………………………………………………………………………………………
(2)
iii)
State two features of fig 1.1 which indicate that the car body is
oscillating in damped harmonic motion.
………Amplitude is decreasing
………follows sine wave (of decreasing amplitude) / has constant period/frequency
………period/frequency is independent of amplitude…………………………
…………………………………………………………………………………………
……………………………………………………………………………………… (2)
b)
Use data from fig 1.1 to calculate the frequency of the car body.
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9 cycles in 6 seconds
6/9 = T
f=1/T
frequency = ………1.5…………….. Hz (2)
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Lesson 28 29 questions – Examples of Resonance
(
/33)………….%..........
ALL
1
The faulty suspension system of a car is tests. The body of the stationary car is
pushed down and released. Fig 1.1 shows how the vertical displacement of the car
varies with time after it has been released.
fig 1.1
a)
i)
The graph shows light damping. Sketch on the same graph
what a critically damped system would look like.
(2)
ii)
Define simple harmonic motion.
………acceleration proportional to displacement
………directed towards a fixed point
…………………………………………………………………………………………
(2)
iii)
State two features of fig 1.1 which indicate that the car body is
oscillating in damped harmonic motion.
………Amplitude is decreasing
………follows sine wave (of decreasing amplitude) / has constant period/frequency
………period/frequency is independent of amplitude…………………………
…………………………………………………………………………………………
……………………………………………………………………………………… (2)
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b)
Use data from fig 1.1 to calculate the frequency of the car body.
9 cycles in 6 seconds
6/9 = T
f=1/T
frequency = ………1.5…………….. Hz
MOST
c)
To simulate the car being driven along a ridged road at different speeds, the
stationary car is oscillated up and down in simple harmonic motion by a mechanical
oscillator. The mechanical oscillator provides a movement of variable frequency and
constant amplitude. Fig 1.2 shows that graph of the vertical motion of the car body
obtained from the test.
0.5
1.0
1.5
2.0
2.5
3.0
fig 1.2
i)
Describe what resonance is.
………Resonance occurs when the driving frequency equals the natural frequency of
an oscillating system.
At this point the amplitude is greater
……………………………………………………………………………………… (3)
ii)
Use information from fig 1.2 to write down the amplitude of
the motion of the mechanical oscillator.
ii)
amplitude = ………5………… mm (1)
Using your answer to (b), add the scale to the frequency axis of
fig 1.2. (1)
iii)
Two new dampers are tried on to increase the damping of the
car body a little and then a heavy damper is fitted. On fig 1.2 sketch the graph you
would expect for these dampers.
(5)
2
In this question, four marks are available for the written communication.
a)
Explain the meaning of the term resonance.
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State two examples of oscillating systems in which resonance occurs; one
being useful or beneficial and the other being a nuisance or harmful. Explain their
practical significance. You may use diagrams in your answer.
State how the oscillation is driven in each case.
……Resonance occurs at/close to the natural frequency of an oscillating system
……caused by a driving force
……when maximum amplitude of driven achieved
………………………………………………………………………………
……good: microwaves, watch, pendulum clock, open and closed pipes, electrical
resonance/tuning
……bad: Tacoma narrows/millennium bridge, wine glass fracture, earthqueake,
motor car wing mirror, steering rattles at different speeds
……………………………………………………………………………………
……practical significance of example stated in a meaningful manner
……nature of driving force clearly stated
……………………………………………………………………………………… (7)
b)
Describe how damping in vibrating systems affects their resonant properties.
Give an example of a practical resonant system where two of the damping effects that
you describe could be observed. A space has been left for you to draw suitable sketch
graph(s), if you wish to illustrate your answer.
without damping
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with damping
………Resonance over a wider range of frequencies
………amplitude decreased
………shift down of resonant frequency with damping
………critiacally/overdamped systems will not vibrate
………………………………………………………………
………Sensible lab demonstration
………realate real system to features
…………………………………………………………………………………………
……………………………………………………………………………………… (5)
Quality of written communication (4)
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