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14. Mass mis whirled on the end of astring length R. The motion isin a vertical plane in the gravitational
field of the earth. The forces on m are the weight W down, and the string force T towards the

centre. The instantaneous speed is v, and the string makes angle © with the horizontal. Find the
T and the tangential acceleration at this instant.

15. A rod of length L has a non-uniform density. 2, the mass per unit length of the rod, varies as
7.5? (6/1), where i) is a constant and s is the distance from the end marked 0. Find the centre
of mass.

16. Apblock of mass M slides along a horizontal table with speed vy. At x = 0, it hits a spring with spring
constant k and begins to experience a friction force, as indicated in the sketch. The co-efficient of

friction is variable and is given by p = bx, where 6 is a constant. Find the distance / the block
travels before coming to rest :

17. A particle of mass m moves in one dimension along the positive x axis. It is acted on by a constant
force directed toward the origin with magnitude B, and an inverse-square law repulsive force with
magnitude A/x?.

(a) Find the potential energy function U (x).

(b) Sketch the energy diagram for the system when the maximum kinetic energy is
Ky = नण

(c) Find the equilibrium position, x».

18. A uniform drum of radius 6 and mass M rolls without slipping down a plane inclined at angle 9,
The moment of inertia of the drum around its axisis Io = Mb?/2. Find the drum’s acceleration along
the plane.

19. Findthe moment of inertia of a thin sheet of mass M in the shape of an equilateral triangle around
an axis through a vertex, perpendicular to the sheet. The length of each side is L.

(Ceiling 30 marks)

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