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MATH172 May 2010 : Mathematics for Physics 2
Examiner: Prof. V. N. Biktashev, Extension 44006.
Time allowed: One hour and a half
You may attempt all questions. All answers to Section A and the best TWO an-
swers to questions from Section B will be takein into account. The marks shown
against questions, or parts of questions, indicate their relative weight. Section A
carries 60% of the available marks.
The vectors
i,
j and
k are unit vectors in the directions of the coordinate axes
Ox, Oy and Oz respectively. Where physical quantities are mentioned, SI units
are implicitly assumed.
SECTION A
- Evaluate matrix products A
2
and AB for matrices A and B given by
A =
, B =
[6 marks]
- Evaluate the vector v = − 2 a + 3
b and its length |v|, if a = 3
i − 2
j +
k
and
b =
j − 3
k.
[4 marks]
- Evaluate the scalar product of a = 3
i − 2
j +
k and
b =
j − 3
k.
[4 marks]
- Evaluate the vector product of a = 3
i − 2
j +
k and
b =
j − 3
k.
[5 marks]
- Find the velocity, speed and acceleration of a particle whose position
vector at time t is
r(t) = 3 cos ti + 4 cos tj + 5 sin t
k.
[5 marks]
- Force
F of magnitude 75 and of the same direction as vector 24
i + 7
j,
moves a particle along a straight line from point A(1, 2) to point B(2, −1). Find
the work done. [6 marks]
- Find the divergence div
E of the vector field
E(x, y, z) = xyzi + sin(x) sin(z)
j + e
y+z k.
[7 marks]
- Find the general solution of the differential equation
dy
dx
= y cos x.
[7 marks]
Paper Code MATH172 May 2010 Page 1 of 4 CONTINUED
- Points A, B, C and D have position vectors
OA =
i+
j+
k,
OB =
i−
j−
k
and
OC = −
i −
j +
k.
Find
AB ×
AC. Hence find
- the area S of triangle ABC,
- two unit vectors n 1 , 2
perpendicular to that triangle,
- the angles α, β, γ of the triangle ABC in degrees, accurate to at least two
decimal places, and verify that the sum of these angles is 180
◦
with the
same precision.
[20 marks]
- A particle moves so that its acceleration at time t is given by
a(t) = 4 sin tj + 4 cos t
k,
and at time t = 0 it was at point (0, 0 , −4) and had velocity v(0) = 3
i − 4
j.
Find the velocity v and the position vector r of this particle at time t = π/2, and
calculate the normal and the tangential components of the acceleration at that
time. [20 marks]
Paper Code MATH172 May 2010 Page 3 of 4 CONTINUED
- A body of mass m is placed on a horizontal surface and attached to a
spring with the Hooke’s spring constant k, as shown on the diagram.
body coordinate
−cx
′
−kx
F (t)
m
x O
The horizontal coordinate x of the body is measured with respect to the equilib-
rium position of the spring. The frictional force acting on the body is proportional
to its velocity with the coefficient c. In addition, the body is affected by a hor-
izontal external force F (t) which is changing with time. The movement of the
body is thus described by a differential equation
m
d
2
x
dt
2
dx
dt
The values of the constants are: m = 1, c = 2, k = 1, and the external force is
F (t) = F 0 cos(ωt), where F 0 = 50 and ω = 3.
Find the general solution x(t) of this differential equation at these values of
parameters.
Hence determine the coordinate x(t) of the body at time t, if at time t = 0 it
was resting at the origin, that is x(0) = 0, dx/dt(0) = 0. In particular, calculate
the coordinate x of the body at the time t = 10π, accurate to five significant
figures. [20 marks]
Paper Code MATH172 May 2010 Page 4 of 4 END
