Marks
6. (i) Masses of 6 lb and 2 lb hang freely at the ends of a light
inelastic string passing over a smooth pulley. Find the
acceleration of the masses and the tension in the string.
The motion starts from rest and, after the lesser mass has
risen 2 ft, it picks up an additional mass of 4 lb. Find the
final velocity of the masses and the tension in the string. (12)
(ii) A mass M moving due East with velocity V collides with
a mass m moving due West with velocity v, the coefficient
of restitution being e. If, after the collision, the mass m
remains at rest, express the ratio ?n : M in terms of e, V, v. (8)
7. State necessary and sufficient conditions for the equilibrium
of a number of coplanar forces acting on a rigid body. (2)
A uniform ladder rests in equilibrium at an angle a to the
horizontal with one end on rough ground and the other
against a rough vertical wall, the ladder being in a vertical plane.
If the coefficients of friction at the lower and upper ends of
the ladder are respectively /x-, and ^2, show that
tan a > . (9)
2^1
If the wall is smooth and /xx = J, and the ladder is at an
angle /3 to the horizontal such that tan /3 == -|, show that the
ladder will slip unless supported, and that this support may
be provided by placing a weight equal to twice that of the
ladder at any point on the lower quarter of the length of the
ladder. (9)
8. A uniform solid consists of the smaller part of a sphere of
radius 4 in, which is cut off by a plane 2 in distant from the
centre of the sphere. Show that the centre of gravity of this
spherical cap is at a distance of 2 • 7 in from the centre of the
sphere. (u
A cylinder of the same material as the sphere and of radius
equal to that of the circular face of the cap is firmly attached
to the cap so that the circular faces coincide. The cylinder is
of height 5 in and the composite solid is placed with its axis
vertical and the curved surface of the cap on a horizontal
surface.
Determine whether the solid is in stable or unstable equilibrium. (H)
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