(a)The vectors A and B are given by:
A=2i +3j +7k
B=-i + 6j -5k
Calculate:-
(i)A×B (ii)A•B (iii) Angle between A and B. (9 marks)
(b)Prove that for any vector function A, the divergence of its curl is zero as in the equation below,
Show that[??•(??×A)]=0 (4 marks)
(c)(i)State the principle of conservation of angular momentum.(1 mark)
(ii)If a particle of mass (m) at a point, whose position vector is (r) moves in a force field F, show that the torque produced is the rate of change in angular momentum.(4 marks)
(d)The depth (h) of a valley in metres is given by;
h(x,y) =3 x2 + 2y2 -50x + xy +20y +300
where y is the distance in meters north and x is the distance in meters east of a tower
Determine:
(i)Co-ordinates of the lowest point on the valley from the tower.
(ii)The depth of the valley . (7 marks)
(e) A body has an initial angular velocity of 3 rad/sec and a constant angular acceleration of 2 rad/sec2. Calculate the
(i)Angular displacement after 3 seconds.
(ii)Angular velocity at t= 3 seconds (5 marks)
QUESTION TWO.
(a)(i)Define a conservative force field.( 1 mark)
(ii)A conservative force may be expressed as:
?(?-F )=- ?(?-? ?(?-V ))
Show that for such a force, ??×F = 0 (4 marks)
(iii)Show that a force field defined by;
F=(6xyz + z2)i + 3 x2yz j + (3 x2y + 2xz)k is a conservative force field. (3 marks)
(iv) Determine the associated potential (V) to the above force field in (a) (iii) (4 marks)
(b) A particle of mass 3 units moves in a force field depending on time t given by;
F=12t2i +(6t – 3)j -3tk .If its position vactor at time t=0 seconds is given by; ro= 2i + j -2 k and its initial velocity at this instant given by; Vo=4i +5 j + 3k
Detemine
Its velocity and position at any time(t).(4 marks)
Torque and angular momentum about the origin for the particle at time(t)
QUESTION THREE.
(i)state TWO conditions for any system of forces acting on an object to be at equilibrium.(2 marks)
(ii)Test whether the following system of forces are in equilibrium.
F1=i –j, acting at a point r1=i + k
F2= i- k, acting at a point r2=2i
F3= 2j + k, acting at a point r3=i-2j
F4=-2i-j , acting at a point r4 =3i + j + k (5 marks)
(b)(i) State the fundamental theorem for gradients.(1 mark)
(ii)For the function; T=x2 + 4 xy + 2 yz3 and the points a=(0, 0, 0), b= (1, 1, 1), check the fundamental theorem for gradients.(5 marks)
(c)(i)Write the mathematical statement for the divergence theorem, explaining the geometrical significance of it. (2 marks)
(ii)Verify divergence theorem for the function;
V=(x2-z2)i+ 2xy j + (y2 + z)k for a unit cube situated at the origin, bounded by the six planes x=y=z=0, x=y=z=1 (5 marks)