(d)
The table below shows the energies of the simple harmonic motion. Complete the
table.
gravitational
potential
energy/J
elastic
potential
energy/J
kinetic
energy/J
total
energy/J
lowest point
equilibrium position
highest point
(e)
On the axes of diagram below, sketch four graphs to show the shape of the variation
with position of the four energies. Label each graph.
[0.785 J, 0.392 J, 3.92 N, 7.00 rad s
-1
, 1.40 m s
-1
, total energy = 1.57 J]
energy/J
lowest point
equilibrium position
highest point
0

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(a)
Frequency is the number of oscillations per unit time and its unit is Hz. Angular
frequency is the product of 2
and frequency and it has a unit of rad s
-1
.
(b)
(i)
Loss in GPE = mge = 0.400 (9.81)0.200 = 0.785 J
(ii)
1
1
1
Gain in EPE =
0.400
9.81 0.200
0.392 J
2
2
2
Te
mge
The difference is the negative work done by the hand exerting an upward force
to gently lower the mass.
(c)
(i)
At lowest point, total extension = 2e
2
=
2
2
mg
T
k
e
e
mg
e
+
Resultant force on the load = 2mg
–
mg = mg = 0.400 (9.81) = 3.92 N upwards
(ii)
o
x
m
ma
F
2
2
-1
3.92
0.400
0.200
7.00 rad s
(iii)
o
x
v
max
-1
s
m
40
.
1
200
.
0
00
.
7
(d)
gravitational
potential
energy/J
elastic
potential
energy/J
kinetic
energy/J
total
energy/J
lowest point
0
1.57
0
1.57
equilibrium position
0.785
0.392
0.392
1.57
highest point
1.57
0
0
1.57
(e)

19. 2013P3Q7 (part)
(a) A tube, sealed at one end, has a uniform area of cross-section
A
. Some sand is placed in
the tube so that it floats upright in a liquid of density
as shown in the figure.
The total mass of the tube and the sand is
m
The tube floats with its base a distance
h
below the surface of the liquid.
Derive an expression relating
m
to
h, A
and
. Explain your working.
(b) The tube is displaced vertically and then released.
For a displacement
x
, the acceleration
a
of the tube is given by the expression
Ag
a
x
m
where
g
is the acceleration of the free fall.
(i)
Explain why the expression leads to the conclusion that the tube is performing
simple harmonic motion.
(ii) The tube has a total mass
m
of 32 g and the area
A
of its cross-section 4.2 cm
2
. It is
floating in liquid of density
of 1.0 x 10
3
kg m
-3
Show that the frequency of oscillation of the tube is 1.8 Hz.
[1.8 Hz]
(a)
Since the tube is in equilibrium, net force = 0.
.
.

(b)(i) Since
, g, A and m are constants, => a
–
x.

2016 Temasek Junior College
12