m the mass of the cylinder, F the initial impulse of friction;

then

u = }u, F=\mu.

(7) Two wheels, revolving uniformly in the same plane, about axes, perpendicular to the plane, through their centres, are suddenly brought into contact, and their axes are kept fixed: to determine what alteration will take place in their angular velocities, the friction being sufficient to prevent all sliding.

1

Let m the mass of one wheel, k = its radius of gyration about its axis of rotation, a = its radius; let a be its angular velocity before and a' after collision. Let n, l, b, B, B', denote like quantities in relation to the other wheel. Then, the revolutions of the two wheels being supposed opposite in character before being brought into contact,

(8) A book ABCD is placed, in a vertical plane, with one angle A on a table: to find the greatest ratio which the side BC can bear to the side AB in order that, after the impact, the book may not tilt over the angle B, the table being supposed to be perfectly rough and the book to be inelastic.

The ratio of BC to AB cannot possibly be greater than

1

√2 how much less the ratio should be is indeterminate, being dependent upon the physical nature of the contact between the side AB of the book and the table.

(9) A spherical ball of given elasticity, moving with a given velocity, and revolving uniformly round a horizontal axis through its centre and perpendicular to the plane of the motion of its centre, impinges upon a horizontal plane of such a nature as to prevent all sliding: to determine whether the angle of reflection from the plane is increased or diminished by increasing the velocity of rotation before impact; and to

find how many revolutions the ball will make after impact before it again strikes the plane.

Let the angles of incidence and reflection be a, a', respectively, and conceive the rotation to be estimated in the direction indicated by the arrows in the diagram: fig. (252): let r= the radius of the ball: let u, v, be the components of the velocity of incidence, parallel and perpendicular to the plane, and angular velocity, the instant before impact.

the

If w, being positive, be increased, a' will increase. If a be negative, or the rotation of an opposite character to that indicated in the figure, then a' will decrease as w increases: if

or the ball will rebound in the normal: if w be a greater nega

tive quantity than

5v 2r

a' will be negative or the angle of

reflection will be on the same side of the normal as the angle of incidence.

The required number of revolutions will be equal to

(10) An inelastic rod rests in a horizontal position on two perfectly rough pegs equidistant from its centre of gravity: if it be turned about one of them, in the vertical plane in which they are situated, and then allowed to fall, to determine whether its motion will cease or not after impact.

Its motion will cease or not as the distance between the pegs is greater or less than

α

√/3

where a is the length of the rod.

Griffin: Solutions of the Examples on the

Motion of a Rigid Body, p. 102.

(11) A perfectly rough plane, moving with a certain velocity parallel to four of the edges of a rigid inelastic cube placed upon it, is suddenly brought to rest to determine the velocity in order that the cube may just turn over its edge.

If c = the length of each edge, and v = the required velocity, v = (√2-1) cg.

(12) A perfectly rough cube rests with one of its faces on a perfectly rough rectangular board, which rests on a smooth horizontal plane, the centre of the base of the cube coinciding with that of the board and the edges of the face being parallel to those of the board: a blow is applied to the board at the middle point of one of its edges in a direction perpendicular to the nearest vertical face of the cube: to find the impulsive stress between the board and the cube, and their motion just after the application of the blow.

Let m be the mass of the cube, m' of the board, B the blow, a the length of an edge of the cube. The horizontal and vertical components of the impulsive stress are equal respectively to

5mB

3mB

5m + 8m2' 5m+8m'

the board's instantaneous velocity is equal to

while the cube revolves for an instant, relatively to the board, about the lower edge nearest the point of impact with an angular velocity equal to

6B

a (5m+ 8m')

(13) A homogeneous sphere, rotating about a horizontal diameter, falls upon a perfectly rough inclined plane through such a height that its angular velocity is not affected by the first impact, and then proceeds to descend the plane directly by bounds to find the velocity of the sphere along the plane just after the nth impact, and to determine the range which the sphere describes upon the plane before it ceases to hop.

Let a be the inclination of the plane to the horizon, h the height through which the sphere falls, e its elasticity: then the required velocity and range are respectively equal to

(14) An imperfectly elastic homogeneous rough sphere is projected obliquely, without rotation, against a fixed plane: to determine P, the ratio of the tangential forces of restitution and compression, in terms of a, a, the angles of incidence and reflection, and e, the coefficient of elasticity for direct impact. The value of p is given by the equation

2p=5-7e tan a' cot a.

Ferrers and Jackson: Solutions of the Cambridge
Froblems, 1848 to 1851.

(15) A series of perfectly rough semicylinders are fixed, side by side, upon their flat faces directly across a straight road of constant inclination: to determine the inclination of the road in order that a rough circular inelastic hoop, just started downwards from the summit of one of the cylindrical ridges, may travel directly along the road with a uniform mean velocity.

=

Let a = the radius of the hoop, a, that of one of the cylinders, the inclination of the road: then, a being given by the formula

=

CHAPTER XIII.

LIVE THINGS.

(1) A FLEA is resting on a needle AB at a given point E: the needle lies on a smooth table: the flea then skips to a given point F of the needle: to determine the least initial velocity of the flea.

Let V be the velocity with which the flea skips, a the inclination of V to the horizon, u the velocity of the needle during the flight of the flea, t the time of flight, m, m', the masses of the needle and flea respectively, and let EF= c.

Then, since the centre of gravity of the flea and needle will not be affected by the skip,

Now the whole range of the flea is equal to the distance EF diminished by the space through which F has slid backwards during the time of flight; and therefore

The least possible value of V is therefore equal to

mcg

(meg