4. A particle of mass m describes a horizontal circle on a smooth cone with vertical axis and vertex upwards, being attached to the vertex by a string of length 1.

If is the velocity of the particle and a the semivertical angle of the cone shew that the pressure of the particle on the cone is

5. Prove that it is necessary and sufficient for the equilibrium of a particle acted on by a number of forces in one plane that the sums of their resolved parts in two directions at right angles should vanish.

A heavy particle of mass m rests on a horizontal plane. To the particle are attached two strings leaving at equal angles a to the plane and in the same vertical plane. These strings pass over smooth pegs and sustain two equal masses M hanging freely. (i.) Find the pressure on the plane. (ii.) If the coefficient of friction between plane and particle is u shew that one of the masses M may be increased by

μ

μ

cos a + μ sin α

without the particle slipping.

6. Shew that a system of forces in one plane, on a rigid body, can in general be reduced to a single force at an arbitrarily chosen point and a couple.

A uniform heavy rod AB of weight w and length 21 rests in contact with a smooth vertical wall at A making an angle a with it in a vertical

plane. The rod rests on a smooth peg at a perpendicular distance h from the wall, and a string attaches B to a point 0 of the wall vertically over A, the angle ABO being a right angle. Shew that the pressure against the wall is w cot a and against the peg

wl (2 sin 2 a)

7. From a body of mass M whose centre of mass is at G, is cut out a piece of mass m whose centre of mass is at G. Find the position of the centre of mass of the remainder.

8. Prove the formula p = gph + II for the pressure in a heavy liquid under atmospheric pressure II at depth h, and define carefully the symbols in

the formula.

Find in dynes the total pressure on the horizontal base of a hemispherical vessel of 1 metre radius full of water, there being a small opening at the top of the vessel, and the barometric height being 76 cm.

9. A body of volume v and sp.g. >1 is supported' wholly immersed, in water, by being attached by a string to a body of volume v' and sp.g. '< 1 which floats partly immersed. Find the volume immersed of the second body and the tension of the string.

10. A circular cylindrical vessel of height h and radius r, closed at the upper end, contains a perfectly fitting piston of weight w and thickness k. The lower face of the piston being flush with the

lower end of the cylinder, and the length h - k of the cylinder containing air at atmospheric pressure II, the cylinder is put vertically into water and pressed down until the lower face of the piston is at a depth 7 below the surface of the water outside. Shew that a length

The Board of Examiners.

1. It is required to hit a point O at a height h above the ground with a particle projected with velocity v from a point on the ground at a horizontal distance c from 0. Shew that the angle of elevation a is given by the equation

gc2 tan2a

2v2c tan a +202h + gc2 = 0.

2. A particle of mass m has a simple rectilinear harmonic motion of period T. Shew that the force on it must be

times the distance from the mean position.

A small smooth heavy ring of mass m is slung on a light elastic string of modulus X and natural length 1, the ends of which are attached at two

fixed points in the same vertical. Shew that the time of a small vertical oscillation is

ml

T

3. Investigate expressions for the velocity and the reaction of a particle on a smooth vertical plane curve under gravity.

A small ring of mass m slides on a smooth vertical circular wire of radius a, and is attached to a light string which passes over a smooth peg at the top 4 of the circle, and, hanging vertically, sustains a particle of mass M. Ifv, v' are the velocities of m and M respectively (the latter moving vertically) when is the angle between Am and the vertical, shew that if the string keeps tight

where Vis the velocity of m at the lowest point of the circle.

4. Prove that if the acceleration of a particle is always directed towards a fixed centre, the particle describes equal areas in equal times around that

centre.

A particle describes an orbit around a centre of force. If the velocity at any point is suddenly increased, shew that a normal force proportional to the curvature at each point will keep it in the same orbit.

5. Prove that in general a disc moving in its own. plane can be brought from one position to any other by rotation around a point properly chosen.

A disc on a plane has one point A jointed to an arm of constant length which can turn about a fixed point B in the plane. A small pin at another fixed point C of the plane can slide in a fine slot in the disc. If AB turns around with angular velocity w, find the instantaneous centre of the disc, its angular velocity, and the velocity of the pin in the slot, any lines or angles to be drawn in the disc being assumed known.

6. A heavy triangular framework ABC formed of bars of weight w per unit length is suspended from C, and weights W1, W, are suspended from A and B respectively, so that AB is horizontal. Shew that

2

{2 W1 + w(b + c)}b cos A

= {2W2+n(a + c)}a cos B, and find the reactions at A.

7. Shew that any system of forces on a rigid body can be reduced to a force R at an arbitrarily chosen point and a couple G.

If the body is constrained to move on a screw of pitch p, whose axis is in the direction of the force R, shew that for equilibrium

where
of G.

PR+ 2TG cos 0 = 0,

is the angle between R and the axis

8. Give a graphical construction for the resultant of a system of forces on a rigid body in one plane.