Department of Mathematics and Philosophy
Faculty of engineering technology
The open university of Sri Lanka
Nawala.
Academic year:
2013/2014
Course Code: MPZ2311
Tutorial. 02
Students need not submit the answers for this tutorial
(01). (a). An uniform rod AB of length a is hung from a smooth peg O by means of a string AOQ of
length l, fastened to the rod at A, and attached at the other end to a light smooth ring Q, which
slides on the rod. Consider the equilibrium of (i) the ring Q, (ii) the rod AB; and deduce that
the straight portions of the string are inclined to the vertical and the rod to the horizontal, at the
same angle , given by acos3 = lsin and, that the tension in the string is
, where w is
the weight of the rod
(b). Two equal uniform rods AB, AC each of length 2b are smoothly jointed together at A, and
rest, in a vertical plane, on a smooth circular cylinder of radius a which is fixed with its axis
horizontal. Show that in the equilibrium position each rod makes with the vertical an angle
given by b sin3 = a cos
(2). (a). A uniform rod of weight W and length 2a is free to move about one end, which is fixed, It is
kept in equilibrium at an inclination 600 to the vertical by a couple which acts in the vertical
plane of the rod. Prove that the moments of the couple
Find the angle between the rod and the vertical when it is in equilibrium under the action of a
couple of half this moment, and prove that the reaction at the fixed end is same in both cases
(b). An uniform bar AB of length 3a and weight W, is freely hinged at A to fixed support. One end
of a light string of length 4a is attached to B and the other end is fixed at C, which is vertically
above A and such tht AC = 3a. Find the tension in the string and the reaction at the hinge in the
position of equilibrium
(03). (a). An uniform rod AB rests with one end B on a rough horizontal plane and is supported by a
string attached to A and to a fixed point, the rod and string being in the same vertical plane. If
and are the angles the string makes with the vertical in the limiting positions of
equilibrium and and the angle the rod makes with the vertical. prove that
Where
is the coefficient of friction;
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(b). A rod AB is limiting equilibrium with A contact with a horizontal floor and B in contact with
a vertical wall, the rod being in a vertical plane perpendicular to the wall. The centre of
gravity of the rod is at G where
and coefficient of friction between the rod and
wall is twice that between the rod and floor. If the rod makes an angle with the wall and the
angle of friction between the rod and floor is . Prove that 4tan = 3tan2
(4) (a). An uniform ladder of weight W rests with the upper end against a smooth vertical wall and the
lower end on a rough horizontal plane, the coefficient of friction being , and the vertical plane
through the ladder being perpendicular to the wall. Prove that no slipping occurs if the inclination
of the ladder to the horizontal is greater than 450.
Find the least inclination of the ladder to the horizontal if a man of weight 3W can ascend to the
top without slipping taking place.
(5).
A
B
F
C
20 kg
D
20 kg
E
10 kg
The above figure represents a framework consisting of seven equal light rods freely jointed of AB, C,
D, E . The frame work carries three loads 20kg, 20kg and 10kg at C, D and E respectively.
It is freely hinged to a fixed point A and is held with CDE horizontal by means of a horizontal force
F applied to the point C. Find F and the horizontal and vertical components of reaction at A. Draw
a stress diagram and hence find the stresses in all the rod distinguishing between tensions and thrusts
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