SJPO Special Round 2010 Sample PDF

SINGAPORE JUNIOR PHYSICS OLYMPIAD 2010
SPECIAL ROUND
4 September, 2010
9:30 am 12:30 pm
Time Allowed: THREE HOURS
INSTRUCTIONS
1. This paper contains 12 structural questions and 7 printed pages.
2. The mark for each question is indicated at the end of the question.
3. Answer ALL the questions in the booklets provided. Answers for Questions 1 6
are to be written in the green booklets provided while answers to Questions 7
12 are to be written in the orange booklets.
4. Scientic calculators are allowed in this test.
5. A table of information is given in page 2.
TABLE OF INFORMATION
Acceleration due to gravity at Earth surface, g = 9.80 m/s
2
Universal gas constant, R = 8.31 J/(mol K)
Vacuum permittivity,
0
= 8.85 10
12
C
2
/(N m
2
)
Vacuum permeability,
0
= 4 10
7
T m/A
Speed of light in vacuum, c = 3.00 10
8
m/s
Speed of sound in air, v = 331 m/s
Charge of electron, e = 1.60 10
19
C
Plancks constant, h = 6.63 10
34
J s
Mass of electron, m
e
= 9.11 10
31
kg
Mass of proton, m
p
= 1.67 10
27
kg
Boltzmann constant, k = 1.38 10
23
J/K
Avogadros number, N
A
= 6.02 10
23
mol
1
Rydberg constant, R
= 1.097 10
7
m
1
Density of air,
air
= 1.20 kg/m
3
Density of gaseous helium,
He
= 0.178 kg/m
3
Standard atmosphere pressure = 1.01 10
5
Pa
Lorentz Transformation: x
=
x ut
1 u
2
/c
2
y
= y
z
= z
t
=
t ux/c
2
1 u
2
/c
2
2
1. A party balloon (of mass 0.0025 kg when empty) is lled with helium to a volume
of 0.0045 m
3
. It is tied to a small stone of mass 0.015 kg by a light string of length
1.5 m to prevent it from ying away. A child holds the balloon at ground level and
then releases it.
(a) How long does it take for the balloon to rise 1.5 m, that is, for the string to
become taut? [ 3 ]
(b) What is the velocity of the stone when it is lifted o the ground? You may
assume that the time needed to bring the stone to this velocity is very short
(impulse approximation) once the string is taut, and the string remains taut
throughout. [ 3 ]
(c) How long after lifting o will the stone touch the ground again? (If you think
that the stone will never touch the ground again, explain why it is so.) [ 3 ]
You may neglect air resistance and treat the stone as a particle.
2. Four blocks are arranged on a smooth horizontal surface as shown. The masses of
the blocks are given (see the diagram). A light rope connects the two top blocks.
The maximum static friction that can exist between each top and the bottom
blocks is f
s
.
(a) What is the maximum value of the horizontal force F, applied to one of the
bottom blocks as shown below, that makes all four blocks move with the same
acceleration? [ 5 ]
(b) If the horizontal force F
is applied to the top block instead, as shown be-

low, what is its maximum value to make all four blocks move with the same
acceleration? [ 3 ]
3
3. (a) A 32-cm-by-32-cm checkerboard has a mass of 100 g. There are four 20-g
checkers located on the checkerboard as shown in the gure. Take the bottom
left corner of the checkerboard to be the origin of our co-ordinate system (0,
0). What are the co-ordinates of the centre of mass of the checkerboard-
checkers system? [ 4 ]
(b) Supposed all the checkers are removed from board, and a circular hole of
radius r = 6 cm with centre at (x, y) = (8, 8) is cut in the board. What are
the co-ordinates of the centre of mass of checkerboard with the hole? [ 5 ]
4. A cone-shaped spaceship of the future uses the solar radiation pressure to propel
itself away from the Sun. The axis of the cone points directly at the Sun as shown.
The conical surface of the ship is painted black. The astronauts then try to increase
their acceleration by covering the conical surface with a highly reective material.
To their dismay, the acceleration actually drops 30%. Find the angle . [ 6 ]
5. Vibrato in a violin is produced by sliding the nger back and forth along the
vibrating string. The G-string on a particular violin measures 30 cm between the
bridge and its far end and is clamped rigidly at both points. Its fundamental
frequency is 197 Hz.
(a) How far from the end should the violinist place a nger so that the G-string
plays the note A (440 Hz)? [ 5 ]
(b) If the violinist executes vibrato by moving the nger 0.50 cm to either side of
the position in part (a), what is the resulting range of frequencies? [ 3 ]
4
6. A vertical cylindrical container contains some helium gas that is in thermodynamic
equilibrium with the surroundings. The gas is conned by a movable heavy piston.
The piston is slowly elevated a distance H from its equilibrium position and then
kept in the elevated position long enough for the thermodynamic equilibrium to be
reestablished. After that, the container is insulated and then the piston is released.
After the piston comes to rest, what is the new equilibrium position of the piston?
[ 8 ]
7. In order to lead an electric current around a right angle in a conducting wire,
surplus charge is required on the outer walls of the wire as shown in the picture
below. The eld lines coming from the left will seem to terminate at the negative
charges at the corner and eld lines going down will seem to originate from the
positive charges.
(a) Assuming that a 1.0 A current exists in a copper wire of 1.0 mm
2
uniform
cross-section, calculate the electric eld that exists in the wire. (Resistivity
of copper is 1.7 10
8
m.) [ 4 ]
(b) Hence, make a good estimate of the amount of charge on each surface at the
right angle turn as indicated. (Hint: the electric eld E is related to the
density of the charges on a surface by E = /
0
) [ 3 ]
8. Find the resistance between points A and B of an innite circuit shown. The
resistance of the resistors in each loop is twice those of the previous loop on its
left.
[ 5 ]
5
9. The following gure shows two circular loops of insulated conducting wires, each of
radius r, is placed in a conguration such that the wires intersect at points A and
B. O and O
denote the respective centers of the wires. In the area of intersection,

a uniform magnetic eld pointing into the paper is applied. The magnetic eld is
dependent linearly on the time, i.e., B = kt where k is a large positive constant.
(a) What is the direction and magnitude of the current in wire loop 1? [ 4 ]
(b) When B = B
0
, what is the force on the loop 1? [ 3 ]
10. (a) A spaceship at rest in a certain reference frame S is given a speed increment
of 0.50c. Relative to its new frame, it is then given a further 0.50c increment.
This process is continued until its speed with respect to its original frame S
exceeds 0.999c. How many increments does this process require? [ 5 ]
(b) The job of the Stanford Linear Accelerator Center (SLAC) in the United
States is to accelerate electrons to very high energies, to crash these electrons
into things and then to see what happens. The acceleration is accomplished
with a 3.2 km long linear accelerator (linac) which is capable of accelerating
electrons such that each electron will have energy of 50 GeV at the exit point
of the accelerator.
i. Assuming that the electrons accelerate constantly down the linac, what
is the speed of the electron after going 1 m down the accelerator? [ 5 ]
ii. When the electrons exit the linear accelerator, magnets are used to curve
the electron beams. Construction plans indicate that the radius of the
curvature is 280 m. By considering the momentum of these now relativis-
tic electrons, what is the minimum magnetic eld required to curve the
electron beams? [ 5 ]
6
11. (a) Liquid helium (atomic weight 4.003) has a density = 0.13 g/cm
3
. Estimate
the radius of a helium atom, assuming that the atoms are packed in the
densest possible conguration, which lls 74% of the space. [ 3 ]
(b) Canal rays, i.e., positive ion rays are generated in a gas discharge tube. How
often does an ion (r = 0.05 nm) collide with an atom of the ideal ller gas
(r = 0.1 nm) if it travels 1 m in a straight path through the discharge tube
and if the pressure in the tube is 1 mbar? 1 mbar corresponds to 10
2
Pa. All
the particles are assumed to have the same velocity. [ 4 ]
12. Astronomers observed a series of spectral lines in the light from a distant galaxy.
On the hypothesis that the lines form the Lyman series for a (new?!) one-electron
atom, they start to construct the energy-level diagram shown in the gure below,
which gives the wavelengths of the rst four lines and the short-wavelength limit
of this series.
(a) Calculate the energies of the ground state and the rst four excited states for
this one-electron atom. [ 3 ]
(b) Calculate the wavelengths of the rst three lines and the short-wavelength
limit in the Balmer series for this atom. [ 2 ]
(c) Show that the wavelengths of the rst four lines and the short-wavelength limit
of the Lyman series for the hydrogen atom are all 60.0% of the wavelengths
for the Lyman series in the one-electron atom described in part (b). [ 3 ]
(d) Based on this observation, explain clearly (with calculation if necessary) why
this atom could be hydrogen. [ 3 ]
End of Paper
7

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SINGAPORE JUNIOR PHYSICS OLYMPIAD 2010

is applied to the top block instead, as shown be-

denote the respective centers of the wires. In the area of intersection,

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