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1PP Examination Autumn 2002
Examiner Dr. S. J. Sweeney
Note: All vector quantities are BOLD FACED and UNDERLINED
Unit vectors are denoted as iˆ , ĵ , k̂
SECTION A: ANSWER ALL QUESTIONS
1.
If the radius of the Earth is R, and the acceleration due to gravity at the surface
is g, what is the acceleration due to gravity at a height


2  1 R above the
Earths surface?
(a) 0.5g
(b) 2g
(c) g
(d)
(e)
2.
g


2 1 g
One object of mass 2kg moves along the x-axis with speed -2vx. Another
object of mass 4kg moves along the x-axis with speed +vx. What is the centreof-mass velocity for this system?
(a) -2vx
(b) 2vx
(c) vx
(d) -vx
(e) 0
3.
Calculate the determinant:
1 2 1
0 2 0
2 2 2
(a) -2
(b) 0
(c) 2
(d) 4
(e) 8
4.
Two forces of ( iˆ + ĵ + k̂ ) N and (2 iˆ -2 ĵ - k̂ ) N act on an object. The unit
vector in the direction of the resultant force is:
(a) (3 iˆ - ĵ ) N
(b)
(c)
(d)
(e)
5.
10 N
1
10
1
2
(3 iˆ - ĵ ) N
(3 iˆ - ĵ ) N
1
10
(-3 iˆ + ĵ ) N
What is the vector product a  b where a = ( iˆ + ĵ - k̂ ) and b = (3 iˆ -2 ĵ + k̂ ):
(a) 3 iˆ +2 ĵ -5 k̂
(b) - iˆ -4 ĵ -5 k̂
(c) - iˆ +4 ĵ -5 k̂
(d) 3 iˆ -4 ĵ -5 k̂
(e) 0
6.
A photon has an energy of 1000meV. This is equivalent to:
(a) 1.6 x 1016 J
(b) 1.6 x 10-16 J
(c) 1.6 x 1019 J
(d) 1.6 J
(e) 1.6 x 10-19 J
7.
An object moves from iˆ m to ( iˆ + ĵ - k̂ ) m whilst being acted upon by a force
( iˆ + ĵ + k̂ ) N. What is the work done by the force on the object?
(a) 0 J
(b) 2 J
(c) ( iˆ + ĵ ) J
(d) 1 J
(e)
8.
5 J
The extension, d, of a spring which obeys Hooke's law is proportional to the
tension, F, which has produced it. Therefore, F=kd. The dimensions of the
constant, k, in terms of mass [M], length [L] and time, [T] are:
(a) [M][L]2[T]-2
(b) [M][T]-2
(c) k is a dimensionless constant
(d) [M]-1[T]2
(e) [L][T]-1
9.
Prior to the metric system, the French unit of length was the toise, where
1 toise = 1949 mm. The toise was divided into 6 Paris feet, each of which was
divided into 12 pouces, that were further subdivided into 12 lignes. How many
cubic lignes occupy the same volume as 1 cubic centimetre?
10.
(a)
8.71x10-2
(b)
0.871
(c)
8.71
(d)
87.1
(e)
871
Which of the following is not an SI unit?
(a) volt
(b) kelvin
(c) mol
(d) metre
(e) candela
11.
A particle moves in a circle with constant angular velocity, ω . If the origin of
co-ordinates is at the centre of the circle, which one of the following
statements concerning the velocity, v , and position, r , of the object is false?
(a) r is constant
(b) r is constant
(c) v is constant
(d) v  ω  r
(e) v  r
12.
If a nucleus is represented by an object the size of a 1p coin, indicate the
distance between two 1p coins that would best correspond to the distance
between two nuclei in a crystal. Assume size of nucleus is 10-15m and typical
spacing of nuclei in a crystal is 10-10m.
13.
(a)
1 cm
(b)
1m
(c)
1 km
(d)
1000 km
(e)
106 km
A particle has a velocity given by v = 3t2 iˆ - ĵ ms-1
The position of the particle as a function of time, given that the particle was at
the origin at t = 2s, is
14.
(a)
t3-t m
(b)
t3 iˆ m
(c)
(t3-8) iˆ + (2-t) ĵ m
(d)
t3 iˆ - t ĵ m
(e)
(t3-8) iˆ m
Which of the following statements about an elastic collision between two
objects is false?
(a)
Some kinetic energy may be converted to heat or sound.
(b)
The two objects cannot bind to form a composite body.
(c)
Total energy is conserved.
(d)
The two objects exert equal and opposite forces on each another.
(e)
The centre-of-mass velocity remains constant.
15.
One light-year (ly) is a unit of length corresponding to the distance that light
travels in one Earth year. To what order of magnitude is the distance from the
Earth to the moon in light years (assume that the speed of light, c=3x108ms-1
and that the moon is 4x105km from Earth).
(a)
10-14 ly
(b)
10-11 ly
(c)
10-8 ly
(d)
10-5 ly
(e)
10-2 ly
(i)
SECTION B: ANSWER TWO QUESTIONS
1.
(a)
The work done by a conservative force, F , on an object moving from P1 to P2
is given by the line integral:
P2
W   F  dr
P1
(i)
(ii)
(iii)
Explain what is meant by a conservative force.
Give one example of a conservative force and one example of a nonconservative force. Briefly explain why they are considered to be
conservative or non-conservative.
If F is not a function of position, show that this equation simplifies to
W  Fr
What does the vector
r represent?
12 marks
(b)
In 1969 the Apollo 11 lunar module (Eagle) successfully landed on the moon.
After completing the mission, the Eagle module then blasted-off the surface to
re-join the command module 15km above the moon's surface.
(i)
(ii)
(iii)
(iv)
Assuming that the mass of the Eagle module is constant during its
ascent and equal to 5000kg (of which 2700kg is fuel) and that the
acceleration due to gravity on the moon, g=1.6ms-1, is assumed to be
constant during the ascent, calculate the work required to lift Eagle to
the command module.
By how much has the gravitational potential energy of Eagle increased
upon reaching the command module?
Given that the radius of the moon, Rmoon=1738km, what is the
percentage change in g on going from the surface to 15km above the
surface?
Is the value calculated for the work done in part (i) likely to be an
over-estimate or under-estimate? Give reasons.
8 marks
2.
(a)
Explain what is meant by,
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
the component of a vector
the magnitude of a vector
the resultant of a set of vectors
a unit vector
If a  5iˆ  3 ˆj  7kˆ and b  3iˆ  ˆj  4kˆ , calculate the unit vector in the
direction of a  b .
What is the smallest angle between vectors a and b ?
Calculate the vector product, a  b .
10 marks
(b)
A particle with charge q and mass m has a velocity v in a region containing a
magnetic field B . The force experienced by the particle is given by
F  qv  B
If the magnetic field has a component in the x direction only, show that the
acceleration of the particle is given by
a

qB ˆ
vz j  v y kˆ
m

where vz and vy are the z and y components of v respectively.
(c)
A particle of mass, m, has a velocity given by the vector equation,

v  Aiˆ  C sin tˆj  costkˆ

where A, C and  are constants. Determine the acceleration of this particle.
(d)
Show that the particle in part (c) has an acceleration and velocity appropriate
to a particle moving in a region containing a uniform magnetic field in the x
direction, as described in part (b). Hence obtain an expression for .
10 marks
3.
(a)
A disc of radius, r, thickness, y, and uniform density, , rotates with an
d
angular velocity, , where  
.
dt
(i)
By considering a sector of the disc, as shown below, develop an
expression for the magnitude of the linear velocity, v, in terms of 
and r.
v
d

dx
r
(ii)
Define the moment of inertia. What does it physically represent?
(iii)
Given that the moment of inertia, I, of a disc is I 
mr 2
, show that the
2
angular momentum, L, of the disc, is given by
L  12 r 4 y
(iv)
For a disc of diameter 1m, thickness 10cm and density 10g/cm3,
calculate its angular momentum if it rotates at 60 revolutions per
minute.
11 marks
(b)
A audio compact disc (CD) consists of an approximately uniform disc of
diameter 120mm, centred on a hole of diameter 15mm. Data is recorded on the
disc from a distance of 24mm from the disc centre to a distance of 58mm from
the disc centre forming a helical track. During playback, the CD is rotated
such that the data is read at a constant linear velocity, v, of 1.3ms-1.
(i)
(ii)
(iii)
Sketch and label a graph showing how the angular velocity, , varies
as a function of the radial distance r such that v is kept constant. Show
the minimum and maximum angular velocities on the graph.
Using the information given above, calculate the total length of the
helical data track for a standard 75 minute audio CD.
To the nearest order of magnitude, estimate the width (pitch) of the
data track on the CD.
9 marks