Topic 13: Oscillations
1
A Level Physics Paper 4 Topical
9702/04/M/J/09/Q4
A vertical peg is attached to the edge of a horizontal disc of radius r, as shown in Fig. 4.1.
peg
disc
r
Fig. 4.1
The disc rotates at constant angular speed ω. A horizontal beam of parallel light produces a
shadow of the peg on a screen, as shown in Fig. 4.2.
screen
peg
R
parallel beam
of light
Q
θ
P
r
S
ω
Fig. 4.2 (plan view)
At time zero, the peg is at P, producing a shadow on the screen at S.
At time t, the disc has rotated through angle θ. The peg is now at R, producing a shadow
at Q.
(a) Determine,
(i)
in terms of ω and t, the angle θ,
............................................................................................................................ [1]
(ii)
in terms of ω, t and r, the distance SQ.
............................................................................................................................ [1]
(b) Use your answer to (a)(ii) to show that the shadow on the screen performs simple
harmonic motion.
..........................................................................................................................................
.................................................................................................................................... [2]
(c) The disc has radius r of 12 cm and is rotating with angular speed ω of 4.7 rad s–1.
Determine, for the shadow on the screen,
(i)
the frequency of oscillation,
(ii)
its maximum speed.
frequency = ......................................... Hz [2]
speed = .................................... cm s–1 [2]
ABDUL HAKEEM
144
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
3
A Level Physics Paper 4 Topical
9702/42/O/N/09/Q3
The variation with displacement x of the acceleration a of the centre of the cone of a
loudspeaker is shown in Fig. 3.1.
750
a / m s–2
500
250
–0.3
–0.2
–0.1
0
0
0.1
0.2
0.3
x / mm
–250
–500
–750
Fig. 3.1
(a)
State the two features of Fig. 3.1 that show that the motion of the cone is simple
harmonic.
1. ......................................................................................................................................
2. ......................................................................................................................................
[2]
(b) Use data from Fig. 3.1 to determine the frequency, in hertz, of vibration of the cone.
frequency = ........................................... Hz [3]
(c) The frequency of vibration of the cone is now reduced to one half of that calculated
in (b). The amplitude of vibration remains unchanged.
On the axes of Fig. 3.1, draw a line to represent the variation with displacement x of the
acceleration a of the centre of the loudspeaker cone.
[2]
ABDUL HAKEEM
146
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
4
A Level Physics Paper 4 Topical
(a) State what is meant by
(i)
9702/41/M/J/10/Q3
oscillations,
..................................................................................................................................
.............................................................................................................................. [1]
(ii)
free oscillations,
..................................................................................................................................
.............................................................................................................................. [1]
(iii)
simple harmonic motion.
..................................................................................................................................
..................................................................................................................................
.............................................................................................................................. [2]
(b) Two inclined planes RA and LA each have the same constant gradient. They meet at
their lower edges, as shown in Fig. 3.1.
ball
L
R
A
Fig. 3.1
A small ball moves from rest down plane RA and then rises up plane LA. It then moves
down plane LA and rises up plane RA to its original height. The motion repeats itself.
State and explain whether the motion of the ball is simple harmonic.
..........................................................................................................................................
..........................................................................................................................................
...................................................................................................................................... [2]
ABDUL HAKEEM
147
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
7
(a)
A Level Physics Paper 4 Topical
9702/41/M/J/11/Q3
Define simple harmonic motion.
..........................................................................................................................................
...................................................................................................................................... [2]
(b) A tube, sealed at one end, has a total mass m and a uniform area of cross-section
The tube floats upright in a liquid of density ρ with length L submerged, as shown in
Fig. 3.1a.
A.
tube
liquid
density ρ
L
L+x
x
Fig. 3.1a
Fig. 3.1b
The tube is displaced vertically and then released. The tube oscillates vertically in the
liquid.
At one time, the displacement is x, as shown in Fig. 3.1b.
Theory shows that the acceleration a of the tube is given by the expression
A ρg
x.
m
Explain how it can be deduced from the expression that the tube is moving with
simple harmonic motion.
a=–
(i)
..................................................................................................................................
.............................................................................................................................. [2]
(ii)
The tube, of area of cross-section 4.5 cm2, is floating in water of density
1.0 × 103 kg m–3.
Calculate the mass of the tube that would give rise to oscillations of frequency 1.5 Hz.
mass = ............................................. g [4]
ABDUL HAKEEM
152
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
9702/41/M/J/12/Q4
9
A small metal ball is suspended from a fixed point by means of a string, as shown in Fig. 4.1.
string
Fig. 4.1
ball
x
The ball is pulled a small distance to one side and then released. The variation with time t of
the horizontal displacement x of the ball is shown in Fig. 4.2.
6
x / cm
4
2
0
0
0.2
0.4
0.6
0.8
t / s / s 1.0
–2
Fig. 4.2
–4
–6
The motion of the ball is simple harmonic.
(a) Use data from Fig. 4.2 to determine the horizontal acceleration of the ball for a
displacement x of 2.0 cm.
acceleration = ....................................... m s–2 [3]
(b) The maximum kinetic energy of the ball is EK.
On the axes of Fig. 4.3, sketch a graph to show the variation with time t of the kinetic
energy of the ball for the first 1.0 s of its motion.
kinetic
energy
EK
0
0
0.2
0.4
0.6
0.8
t /s
1.0
Fig. 4.3
[3]
ABDUL HAKEEM
154
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
9702/42/O/N/12/Q4
10
A ball is held between two fixed points A and B by means of two stretched springs, as shown
in Fig. 4.1.
ball
A
B
Fig. 4.1
The ball is free to oscillate horizontally along the line AB. During the oscillations, the springs
remain stretched and do not exceed their limits of proportionality.
The variation of the acceleration a of the ball with its displacement x from its equilibrium
position is shown in Fig. 4.2.
15
a / m s–2
10
5
–3
–2
–1
0
0
1
2
3
x / cm
–5
–10
–15
Fig. 4.2
ABDUL HAKEEM
155
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
(a)
A Level Physics Paper 4 Topical
State and explain the features of Fig. 4.2 that indicate that the motion of the ball is
simple harmonic.
..........................................................................................................................................
..........................................................................................................................................
..........................................................................................................................................
..........................................................................................................................................
...................................................................................................................................... [4]
(b) Use Fig. 4.2 to determine, for the oscillations of the ball,
(i)
the amplitude,
amplitude = .......................................... cm [1]
(ii)
the frequency.
frequency = ........................................... Hz [3]
(c) The arrangement in Fig. 4.1 is now rotated through 90° so that the line AB is vertical.
The ball now oscillates in a vertical plane.
Suggest one reason why the oscillations may no longer be simple harmonic.
..........................................................................................................................................
...................................................................................................................................... [1]
ABDUL HAKEEM
156
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
9702/41/M/J/13/Q3
12
A ball is held between two fixed points A and B by means of two stretched springs, as shown
in Fig. 3.1.
A
B
ball
Fig. 3.1
The ball is free to oscillate along the straight line AB. The springs remain stretched and the
motion of the ball is simple harmonic.
The variation with time t of the displacement x of the ball from its equilibrium position is
shown in Fig. 3.2.
2.0
x / cm
1.0
0
0
0.2
0.6
0.4
0.8
1.0
t /s
1.2
–1.0
–2.0
Fig. 3.2
(a)
(i)
Use Fig. 3.2 to determine, for the oscillations of the ball,
1.
the amplitude,
amplitude = ........................................... cm [1]
2.
the frequency.
frequency = ............................................ Hz [2]
ABDUL HAKEEM
159
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
(ii) Show that the maximum acceleration of the ball is 5.2 m s–2.
[2]
(b) Use your answers in (a) to plot, on the axes of Fig. 3.3, the variation with displacement x
of the acceleration a of the ball.
a / m s–2
0
0
x / 10–2 m
Fig. 3.3
[2]
(c) Calculate the displacement of the ball at which its kinetic energy is equal to one half of
the maximum kinetic energy.
displacement = ........................................... cm [3]
ABDUL HAKEEM
160
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
16
A Level Physics Paper 4 Topical
9702/42/O/N/14/Q4
(a) State what is meant by simple harmonic motion.
...................................................................................................................................................
.............................................................................................................................................. [2]
(b) A small ball rests at point P on a curved track of radius r, as shown in Fig. 4.1.
curved track,
radius r
x
Fig. 4.1
P
The ball is moved a small distance to one side and is then released. The horizontal
displacement x of the ball is related to its acceleration a towards P by the expression
gx
a = −
r
where g is the acceleration of free fall.
(i)
Show that the ball undergoes simple harmonic motion.
...........................................................................................................................................
...................................................................................................................................... [2]
(ii)
The radius r of curvature of the track is 28 cm.
Determine the time interval τ between the ball passing point P and then returning to
point P.
τ = ..................................................... s [3]
(c) The variation with time t of the displacement x of the ball in (b) is shown in Fig. 4.2.
x
0
0
o
2ot
3ot
4 ot
t
Fig. 4.2
Some moisture now forms on the track, causing the ball to come to rest after approximately
15 oscillations.
On the axes of Fig. 4.2, sketch the variation with time t of the displacement x of the ball for
the first two periods after the moisture has formed. Assume the moisture forms at time t = 0.
[3]
ABDUL HAKEEM
165
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
9702/43/O/N/14/Q4
18 (a) State what is meant by simple harmonic motion.
...................................................................................................................................................
...................................................................................................................................................
.............................................................................................................................................. [2]
(b) A trolley is attached to two extended springs, as shown in Fig. 4.1.
spring
trolley
Fig. 4.1
The trolley is displaced along the line joining the two springs and is then released. At one
point in the motion, a stopwatch is started. The variation with time t of the velocity v of the
trolley is shown in Fig. 4.2.
v
0
0
0.5
1.0
1.5
t/s
2.0
Fig. 4.2
The motion of the trolley is simple harmonic.
(i)
State one time at which the trolley is moving through the equilibrium position and also
state the next time that it moves through this position.
.............................................. s and .............................................. s [1]
ABDUL HAKEEM
168
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
(ii)
A Level Physics Paper 4 Topical
The amplitude of vibration of the trolley is 3.2 cm.
Determine
1. the maximum speed v0 of the trolley,
v0 = ............................................. cm s−1 [3]
2. the displacement of the trolley for a speed of ½v0.
displacement = .................................................. cm [2]
(c) Use your answers in (b) to sketch, on the axes of Fig. 4.3, a graph to show the variation with
displacement x of the velocity v of the trolley.
v
0
–4
–2
0
2
x / cm 4
Fig. 4.3
[2]
ABDUL HAKEEM
169
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
20
A Level Physics Paper 4 Topical
9702/42/M/J/15/Q4
(a) State what is meant by simple harmonic motion.
...................................................................................................................................................
...................................................................................................................................................
...............................................................................................................................................[2]
(b) The variation with time t of the displacement x of two oscillators P and Q is shown in Fig. 4.1.
4
3
x / cm
2
oscillator P
oscillator Q
1
0
0
0.4
0.8
1.2
1.6
2.0
2.4
t/s
2.8
−1
−2
−3
−4
Fig. 4.1
The two oscillators each have the same mass.
Use Fig. 4.1 to determine
(i)
the phase difference between the two oscillators,
phase difference = ................................................... rad [1]
(ii)
the maximum acceleration of oscillator Q,
maximum acceleration = ................................................ m s–2 [2]
ABDUL HAKEEM
171
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
(iii)
A Level Physics Paper 4 Topical
the ratio
maximum kinetic energy of oscillations of Q
maximum kinetic energy of oscillations of P .
ratio = .......................................................... [2]
(c) Use data from (b) to sketch, on the axes of Fig. 4.2, the variation with displacement x of the
acceleration a of oscillator Q.
a
–4
–3
–2
–1
0
0
1
2
3
4
x / cm
Fig. 4.2
[2]
ABDUL HAKEEM
172
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
24
A Level Physics Paper 4 Topical
9702/43/O/N/16/Q3
To demonstrate simple harmonic motion, a student attaches a trolley to two similar stretched
springs, as shown in Fig. 3.1.
spring
trolley
Fig. 3.1
The trolley has mass m of 810 g.
The trolley is displaced along the line of the two springs and then released. The subsequent
acceleration a of the trolley is given by the expression
a = −
2k x
m
where the spring constant k for each of the springs is 64 N m−1 and x is the displacement of the
trolley.
(a) Show that the frequency of oscillation of the trolley is 2.0 Hz.
(b) The maximum displacement of the trolley is 1.6 cm.
Calculate the maximum speed of the trolley.
[3]
speed = ............................................... m s−1 [2]
(c) The mass of the trolley is increased. The initial displacement of the trolley remains unchanged.
Suggest the change, if any, that occurs in the frequency and in the maximum speed of the
oscillations of the trolley.
frequency: .................................................................................................................................
maximum speed: ......................................................................................................................
[2]
178
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
9702/42/O/N/16/Q4
25 A mass hangs vertically from a fixed point by means of a spring, as shown in Fig. 4.1.
spring
l
mass
Fig. 4.1
The mass is displaced vertically and then released. The subsequent oscillations of the mass are
simple harmonic.
The variation with time t of the length l of the spring is shown in Fig. 4.2.
18
17
l / cm
16
15
14
13
12
0
0.1
0.2
0.3
Fig. 4.2
(a)
0.4
0.5
0.6
t /s
Use Fig. 4.2 to
(i)
state two values of t at which the mass is moving downwards with maximum speed,
t = ................................. s and t = ................................. s [1]
179
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
(ii)
A Level Physics Paper 4 Topical
determine, for these oscillations, the angular frequency ω,
ω = .............................................. rad s–1 [2]
(iii)
show that the maximum speed of the mass is 0.42 m s–1.
[2]
(b) Use data from Fig. 4.2 and (a)(iii) to sketch, on the axes of Fig. 4.3, the variation with
displacement x from the equilibrium position of the velocity v of the mass.
0.5
0.4
v / m s–1
0.3
0.2
0.1
0
–4
–3
–2
–1
0
–0.1
1
2
3
4
x / cm
–0.2
–0.3
–0.4
–0.5
Fig. 4.3
180
[3]
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
30 (a) (i)
A Level Physics Paper 4 Topical
Define the radian.
9702/42/O/N/17/Q3
...........................................................................................................................................
...........................................................................................................................................
.......................................................................................................................................[2]
(ii)
State, by reference to simple harmonic motion, what is meant by angular frequency.
...........................................................................................................................................
.......................................................................................................................................[1]
(b) A thin metal strip, clamped horizontally at one end, has a load of mass M attached to its free
end, as shown in Fig. 3.1.
clamp
L
x
metal strip
Fig. 3.1
oscillation
of load
load
mass M
The metal strip bends, as shown in Fig. 3.1.
When the free end of the strip is displaced vertically and then released, the mass oscillates in
a vertical plane.
Theory predicts that the variation of the acceleration a of the oscillating load with the
c
displacement x from its equilibrium position is given by a = – c 3 m x
ML
where L is the effective length of the metal strip and c is a positive constant.
ABDUL HAKEEM
188
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
(i)
A Level Physics Paper 4 Topical
Explain how the expression shows that the load is undergoing simple harmonic motion.
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
.......................................................................................................................................[2]
(ii)
For a metal strip of length L = 65 cm and a load of mass M = 240 g, the frequency of
oscillation is 3.2 Hz.
Calculate the constant c.
c = ........................................... kg m3 s–2 [3]
31
9702/42/F/M/18/Q3(a)
A mass is undergoing simple harmonic motion with amplitude x0. The maximum velocity of
the mass has magnitude v0.
On Fig. 3.1, show the variation with displacement x of the velocity v of the mass.
v
v0
−x0
0
x0
0
x
−v0
Fig. 3.1
189
[2]
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
9702/41/M/J/18/Q2
33 A metal plate is made to vibrate vertically by means of an oscillator, as shown in Fig. 2.1.
sand
plate
direction of
oscillations
oscillator
Fig. 2.1
Some sand is sprinkled on to the plate.
The variation with displacement y of the acceleration a of the sand on the plate is shown in
Fig. 2.2.
5
a / m s–2
4
3
2
1
–10
–8
–6
–4
–2
0
0
–1
2
4
6
8
10
y / mm
–2
–3
–4
–5
Fig. 2.2
191
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
(a) (i)
A Level Physics Paper 4 Topical
Use Fig. 2.2 to show how it can be deduced that the sand is undergoing simple harmonic
motion.
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
.......................................................................................................................................[2]
(ii)
Calculate the frequency of oscillation of the sand.
frequency = ..................................................... Hz [2]
(b) The amplitude of oscillation of the plate is gradually increased beyond 8 mm. The frequency
is constant.
At one amplitude, the sand is seen to lose contact with the plate.
For the plate when the sand first loses contact with the plate,
(i)
state the position of the plate,
.......................................................................................................................................[1]
(ii)
calculate the amplitude of oscillation.
amplitude = ................................................... mm [3]
192
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
9702/42/M/J/18/Q4
34
(a) State two conditions necessary for a mass to be undergoing simple harmonic motion.
1. ...............................................................................................................................................
...................................................................................................................................................
2. ...............................................................................................................................................
...................................................................................................................................................
[2]
(b) A trolley of mass 950 g is held on a horizontal surface by means of two springs attached to
fixed points P and Q, as shown in Fig. 4.1.
trolley
mass 950 g
spring
P
Q
Fig. 4.1
The springs, each having a spring constant k of 230 N m–1, are always extended.
The trolley is displaced along the line of the springs and then released.
The variation with time t of the displacement x of the trolley is shown in Fig. 4.2.
x
0
0
t1
t
Fig. 4.2
193
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
(i)
1.
A Level Physics Paper 4 Topical
State and explain whether the oscillations of the trolley are heavily damped, critically
damped or lightly damped.
....................................................................................................................................
....................................................................................................................................
2.
Suggest the cause of the damping.
....................................................................................................................................
....................................................................................................................................
....................................................................................................................................
[3]
(ii)
The acceleration a of the trolley of mass m may be assumed to be given by the expression
a=–d
1.
2k
nx .
m
Calculate the angular frequency ω of the oscillations of the trolley.
ω = ............................................... rad s–1 [3]
2.
Determine the time t1 shown on Fig. 4.2.
t1 = ....................................................... s [2]
194
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
ANSWERS
1 (a)
(i) (θ =) ω t (allow any subject if all terms given)
(SQ =) r sinωt (allow any subject if all terms given)
(b) this is the solution of the equation a = –ω2x
a = –ω2x is the (defining) equation of s.h.m.
(c) (i) f =
=
=
(ii) v =
=
=
3
A1 [2]
ω / 2π
4.7 / 2π
0.75 Hz
rω (r must be identified)
4.7 × 12
56 cm s–1
C1
A1 [2]
C1
A1 [2]
(a) straight line through origin ......................................................................................... B1
negative gradient ....................................................................................................... B1 [2]
(b) a = -ω2x and ω = 2πf .......................................................................................... C1
750 = (2πf)2 × 0.3 × 10-3 ........................................................................................... C1
f = 250 Hz ................................................................................................................
A1 [3]
(c) straight line between(-0.3,+190) and (+0.3,-190) ...................................................... A2 [2]
(allow 1 mark for end of line incorrect by one grid square or line does not extend to
+/- 0.3 mm)
4
(a) (i) to-and-fro / backward and forward motion (between two limits)
B1 [1]
(ii) no energy loss or gain / no external force acting / constant energy / constant amplitude
B1 [1]
(iii) acceleration directed towards a fixed point
B1
acceleration proportional to distance from the fixed point / displacement
B1 [2]
(b) acceleration is constant (magnitude)
so cannot be s.h.m.
M1
A1
195
[2]
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
ANSWERS
7
(a) acceleration / force proportional to displacement from a fixed point
acceleration / force (always) directed towards that fixed point / in opposite
direction to displacement
(b) (i) Aρg / m is a constant and so acceleration proportional to x
negative sign shows acceleration towards a fixed point / in opposite
direction to displacement
2
(ii) ω = (Aρg / m)
ω = 2πf
(2 × π × 1.5)2 = ({4.5 × 10–4 × 1.0 × 103 × 9.81} / m)
m = 50 g
196
M1
A1 [2]
B1
B1 [2]
C1
C1
C1
A1 [4]
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
ANSWERS
9
(a) a = (–)ω2x and ω = 2π/T
T = 0.60 s
a = (4π2 × 2.0 × 10–2) / (0.6)2
= 2.2 m s–2
(b) sinusoidal wave with all values positive
10
C1
C1
A1
[3]
B1
all values positive, all peaks at EK and energy = 0 at t = 0
B1
period = 0.30 s
B1
[3]
(a) straight line through origin
shows acceleration proportional to displacement
negative gradient
shows acceleration and displacement in opposite directions
M1
A1
M1 [4]
A1
(b) (i) 2.8 cm
(ii) either gradient = ω2 and ω = 2πf or a = –ω2x and ω = 2πf
A1
C1
gradient = 13.5 / (2.8 × 10–2) = 482
ω = 22 rad s–1
π
frequency = (22/2 =) 3.5 Hz
[1]
C1 [3]
A1
(c) e.g. lower spring may not be extended
e.g. upper spring may exceed limit of proportionality / elastic limit
(any sensible suggestion)
[1]
B1
Topic 13: Oscillations
A Level Physics Paper 4 Topical
ANSWERS
12 (a)
(i) 1.
2.
amplitude = 1.7 cm
A1 [1]
period
= 0.36 cm
frequency = 1/0.36
frequency = 2.8 Hz
C1
(ii) a = (–)ω2x and ω = 2π/T
acceleration = (2π/0.36)2 × 1.7 × 10–2
= 5.2 m s–2
(b) graph:
straight line, through origin, with negative gradient
from (–1.7 × 10–2, 5.2) to (1.7 × 10–2, –5.2)
(if scale not reasonable, do not allow second mark)
(c) either kinetic energy = ½mω2(x02 – x2)
or
potential energy = ½mω2x2 and potential energy = kinetic energy
2
½mω (x0 – x2) = ½ × ½mω2x02 or ½mω2x2 = ½ × ½mω2x02
x02 = 2x2
x = x0 / √2 = 1.7 / √2
= 1.2 cm
A1 [2]
C1
M1
A0
[2]
M1
A1 [2]
B1
C1
A1
[3]
Topic 13: Oscillations
A Level Physics Paper 4 Topical
ANSWERS
16 (a) acceleration / force proportional to displacement (from a fixed point)
either acceleration and displacement in opposite directions
or acceleration always directed towards a fixed point
(b) (i) g and r are constant so a is proportional to x
negative sign shows a and x are in opposite directions
(ii) ω 2 = g / r and ω = 2π / T
ω 2 = 9.8 / 0.28
= 35
T = 2π / √35 = 1.06 s
time interval τ = 0.53 s
(c) sketch: time period constant (or increases very slightly)
drawn line always ‘inside’ given loops
successive decrease in peak height
M1
A1 [2]
B1
B1 [2]
C1
C1
A1 [3]
M1
A1
A1 [3]
Topic 13: Oscillations
A Level Physics Paper 4 Topical
ANSWERS
18
(a)
acceleration / force proportional to displacement (from a fixed point)
either acceleration and displacement in opposite directions
or acceleration always directed towards a fixed point
(b) (i) zero & 0.625 s or 0.625 s & 1.25 s or 1.25 s & 1.875 s or 1.875 s & 2.5 s
(ii) 1.
ω = 2π / T and v0 = ωx0
ω = 2π / 1.25
= 5.03 rad s–1
A1 [1]
C1
A1 [3]
v = ω ( x 02 − x 2 )
either
½ωa = ω ( x 02 − x 2 )
2
2
x0 / 4 = x 0 – x
x = 2.8 cm
20
A1 [2]
C1
v0 = 5.03 × 3.2
= 16.1 cm s–1 (allow 2 s.f.)
2.
M1
or
2
½ × 16 .1 = 5.03 (3.22 − x 2 )
2
2.58 = 3.2 – x
x = 2.8 cm
C1
2
A1 [2]
(c) sketch: loop with origin at its centre
correct intercepts & shape based on (b)(ii)
M1
A1 [2]
(a) displacement (directly) proportional to acceleration/force
either displacement and acceleration in opposite directions
acceleration (always) towards a (fixed) point
or
M1
(b) (i) ⅓π rad or 1.05 rad (allow 60° if unit clear)
(ii) a0 = –ω2 x0
= (–) (2π / 1.2)2 × 0.030
= (–) 0.82 m s–2
(special case: using oscillator P gives x0 = 1.7 cm and a0 = 0.47 m s–1 for 1/2)
(iii) max. energy ∝ x02
ratio = 3.02 / 1.72
= 3.1 (at least 2 s.f.)
(if has inverse ratio but has stated max. energy ∝ x02 then allow 1/2)
(c) graph: straight line through (0,0) with negative gradient
correct end-points (–3.0, +0.82) and (+3.0, –0.82)
200
A1 [2]
A1 [1]
C1
A1 [2]
C1
A1 [2]
M1
A1 [2]
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
ANSWERS
24 (a) 2k/m = ω2
ω = 2πf
(2 × 64 / 0.810) = (2π × f)2 leading to f = 2.0 Hz
(b) v0 = ωx0
or
A1
M1
M1
[3]
v0 = 2πfx0
or v = ω(x02 – x2)1/2 and x = 0
v0 = 2π × 2.0 × 1.6 × 10
C1
–2
= 0.20 m s–1
(c) frequency: reduced/decreased
maximum speed: reduced/decreased
201
A1
[2]
B1
B1
[2]
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
ANSWERS
25
(a) (i) 0.225 s and 0.525 s
(ii) period or T = 0.30 s and ω = 2π / T
ω = 2π / 0.30
ω = 21 rad s–1
(iii) speed = ωx0 or ω(x02 – x2)1/2 and x = 0
= 20.9 × 2.0 × 10–2 = 0.42 m s–1
or use of tangent method:
correct tangent shown on Fig. 4.2
working e.g. ∆y / ∆x leading to maximum speed in range (0.38–0.46) m s–1
(b) sketch: reasonably shaped continuous oval/circle surrounding (0,0)
curve passes through (0, 0.42) and (0, –0.42)
curve passes through (2.0, 0) and (–2.0, 0)
A1 [1]
C1
A1 [2]
C1
A1 [2]
(C1)
(A1)
B1
B1
B1 [3]
Topic 13: Oscillations
A Level Physics Paper 4 Topical
ANSWERS
30 (a) (i)
angle (subtended) where arc (length) is equal to radius
(angle subtended) at the centre of a circle
(ii) angular frequency = 2π × frequency or 2π / period
(b) (i)
B1
c / ML3 is a constant so acceleration is proportional to displacement
minus sign shows that acceleration and displacement are in opposite directions
3
2
(ii) c / ML = (2πf )
c = 4π2 × 3.22 × 0.24 × 0.653
M1
A1
B1
B1
C1
C1
= 27 kg m3 s–2
A1
31 reasonably shaped circle or oval surrounding the origin
closed loop passing through (0,±v0) and (±x0,0)
203
B1
B1
Airport Road / JoharTown /BahriaTown
Topic 13: Oscillations
A Level Physics Paper 4 Topical
ANSWERS
33 (a) (i) straight line through origin indicates acceleration ∝ displacement
negative gradient shows acceleration and displacement are in opposite directions
(ii) a = –ω2y and ω = 2πf
4.5 = (2π × f)2 × 8.0 × 10–3 (or other valid read-off)
f = 3.8 Hz
(b) (i) maximum displacement upwards/above rest/above the equilibrium position
(ii) (just leaves plate when) acceleration = 9.81 m2s–
or 9.81 = 563 × y0
9.81 = (2π × 3.8)2 × y0
amplitude = 17 mm
34 (a)
acceleration proportional to displacement
acceleration directed towards fixed point
or displacement and acceleration in opposite directions
(b) (i) 1. amplitude decreases gradually so light damping
or oscillations continue so light damping
2. loss of energy
due to friction in wheels
or due to friction between wheels and surface (during slipping)
or due to air resistance (on trolley)
(ii) 1
ω2 = 2k / m
A1
B1
C1
C1
A1
B1
B1
B1
B1
B1
C1
C1
= (2 × 230) / 0.950
2
B1
B1
C1
ω
= 22 rad s–1
T = 2π / ω
T = (2π / 22) = 0.286 s
time = 1.5T
= 0.43 s
A1
C1
A1
204
Airport Road / JoharTown /BahriaTown