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Math 113: Vectors And Mechanics Question Paper
Math 113: Vectors And Mechanics
Course:Bachelor Of Education (Science & Arts)
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
FIRST YEAR EXAMINATION FOR THE AWARD OF DEGREE OF
BACHELOR OF EDUCATION (SCIENCE & ARTS)
MATH 113: VECTORS AND MECHANICS
STREAMS: BED (SCI) YIS2 & BED (ARTS) YIS2 TIME: 2 HOURS
DAY/DATE: WEDNESDAY 14/8/2013 11.30 AM – 1.30 PM
INSTRUCTIONS:
Answer Question One (Compulsory) and any other Two Questions
Adhere to the instructions on the answer booklet.
Question One (Compulsory) (30 Marks)
(a) Define the following terms:
(i) Coplanar vectors
(ii) Moment of a force
(iii) Unit vector [3 marks]
(b) Given (AB) ?=a ? and (AC) ?=b ?, show that the area of the triangle ABC is given by
Area = 1/2 v((ab)^2-(a ?· b ? )^2 ) [4 marks]
(c) Given a ? and b ? are two non – parallel vectors, show that vector addition is distributive. [3 marks]
(d) Find the angle between vectors
a ?=3i ?+7j ?-2k ?
b ?=11j ?-8k ? [4 marks]
(e) Find the parametric and Cartesian equation of a line that is parallel to the vector
2i ?+3j ?-4k ? and passes through a point whose position vector is 3i ?-j ?+2k ?.
[3 marks]
(f) A force (3i ?-2j ?+2k ? )N and a force (7i ?+4j ?-6k ? )N act on a particle during a displacement (5i ?-6j ?+k ? ) metres. Calculate the work done. [4marks]
(g) A particle moving in a straight line with constant acceleration travels 10m in the first second and 15 m in the second second. Find the distance travelled in the third second.
[5 marks]
(h) Given A ?=x^2 yi ?+y^2 zj ?+z^2 xk ? determine ? ?.A ? at (1,-2,3). [4 marks]
Question Two (20 Marks
(a) (i) State without proof Lamis Theorem of forces acting at a point [2 marks]
(ii) the resultant of the two forces 2F and 3F acting at a point is v7 F. Find:
(I) The angle between the two forces. [4 marks]
(II) The direction of the resultant to the horizontal [3 marks]
(III) The line of action of the resultant [1 mark]
(b) A uniform ladder of man 30kg and length 4m stands on a rough horizontal ground and leans against a smooth vertical wall. The foot of the ladder is 1.2 m form the wall. Find the normal contact force of the wall on the ladder and of the ground on the ladder. Find also the frictional force of the ground on the ladder (Take g=9.8 ?ms?^(-2) ) [10 marks]
Question Three (20 Marks)
(a) Given a ?·b ?=a ?·c ?, show that a ? is perpendicular to b ?-c ? [3 marks]
(b) In the fig below, the particle of mass 2 kg is held in equilibrium on a smooth plane of angle 30° to the horizontal by the force FN acting at an angle of 30° to the plane.
FN
30°
2kg
30°
Find the value of F and the normal reaction of the plane on the particle. [4 marks]
(c) A weight W hangs from a fixed point O by an inextensible string. The weight is pushed aside by a horizontal force F and rests in equilibrium with the string inclined at an angle 60° to the vertical.
Determine:
(i) The horizontal force F in terms of W
(ii) The tension in the string in terms of W. [3 marks]
(d) Find the area of triangle ABC if co-ordinates of A, B and C are (3,2,1),(5,-3,2),(-1,2,0) respectively. [4 marks]
(e) Given a vector r ?=4i ?+3j ?-k ?, find the unit vector in the direction of r ?. [3 marks]
Question Four (20 Marks)
(a) The figure below shows a body A of mass 2.6 kg on a rough inclined plane PQR connected by a light inextensible string passing over a smooth pulley S to a body B of mass 5.4kg hanging freely. PQ=13m,QR=5m,AQ=4m and QB=2m. The coefficient of friction between A and the plane is ½. When the bodies are released from rest, calculate:
(I) The common acceleration of the system. After B has descended through 2 m, the string breaks.
(II) Calculate the speed of B at that instant.
(III) Calculate the time from the start for B to strike the ground.
(IV) How much further up the plane will A move?
S
Q
A 5m
P R
(b) Find the perpendicular distance between the point p(4,-3,10) and the line whose
vector equation is r ?=(¦(1@2@3))+x(¦(3@-1@2)) [8 marks]
Question Five (20 Marks)
(a) Find the point where the line r ?=(-i) ?-3j ?+4k ?+?(2i ?-j ?-3k ? ) intersects the plane
r ?·(i ?-j ?+2k ? )=-5 [5 marks]
(b) Show that the forces of magnitude 40N acting at the vertices of the equilateral triangle
ABC and parallel to the opposite sides form a couple. Find the magnitude of the couple
if each side of the triangle is 50 cm long.
B 40 N
y
40N
x
0
A C
40N
(c) Given that A ?=x^2 i ?+y^2 zj ?+z^2 xk ? . Find
(i) ? ?×A ? [3 marks]
(ii) ? ?·A ? [3 marks]
(d) Given Ø=e^(x^2+y^2+z^2 ), determine Grad Ø. [3 marks]
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