The required initial velocities are in the proportion of 2, 5, 2, -1.

(19) A rectangle, formed of four uniform rods, which are connected by hinges at their ends, is revolving about its centre on a smooth horizontal plane with a given angular velocity, when a point in one of the sides suddenly becomes fixed: to find the angular velocity of either of the sides adjacent to the side with the fixed point, immediately after it becomes fixed.

If 2a be the length of the side of which a point becomes fixed, 2b the length of an adjacent side, and ∞ the given angular velocity of the rectangle, the required angular velocity is equal to

(20) A perfectly inelastic and smooth ellipsoid the semiaxes of which are a, b, c, revolving with an angular velocity w round one axis c, impinges with a velocity v upon a quiescent sphere of equal magnitude: the instant before collision, the semi-axis a lies in the direction of the motion of the centre of gravity of the ellipsoid: at the instant of impact, the sphere touches the ellipsoid at the extremity of the latus rectum of its principal section containing a and b: supposing the eccentricity of that principal section to be equal to √, to determine he relation between v and w in order that there may be no rotatory motion in the ellipsoid after collision.

The required relation is

= 2a.

3

SECT. 4. Rough Surfaces.

(1) An inelastic cylinder O (fig. 248) having rolled down a perfectly rough plane CA, impinges upon a perfectly rough plane CA, the axis of the cylinder being parallel to the

intersection of the two planes: to find the velocity with which the cylinder will commence its ascent up the second plane, and the limiting angle of inclination of the two planes for which the ascent is possible.

Let CAC'a, k = the radius of gyration of the cylinder about its axis, a = the radius of the cylinder, m = its mass; u = the velocity of the centre O of a circular section of the cylinder just before impact, and the velocity after impact up the plane AC'; R= the impulsive force of friction exerted by the plane AC' upon the cylinder at the moment of impact to secure perfect rolling.

Then, for the motion of the centre of gravity of the cylinder parallel to AC', we have

and, for the value of the decrement of the angular velocity of the cylinder about its axis owing to the impulse R, we have the expression

which, by (1), is equal to

Ra mk2

a (v + u cos a)
k

but, the planes being both perfectly rough, it is evident that

the angular velocity before, and after the impact: hence

a

2= (v + u cos a):
k*

u-v=2v + 2u cos a ;

v = } (1 − 2 cos a) u,

is

a

which gives the velocity with which the cylinder begins to ascend the plane AC'.

Since, from the nature of the case, v cannot have a negative value, it is clear that the ascent is impossible unless a be greater than π.

(2) An inelastic cylinder rolls without sliding along a plane, and impinges upon a perfectly rough fixed point, the circular section of the cylinder through the rough point being supposed to bisect the axis of the cylinder: to determine the least distance of the point from the plane in order that the cylinder may be reduced to rest by the impact.

Let o be the angular velocity of the cylinder just before and w' just after impact; the cylinder being supposed to turn over the fixed point C, (fig. 249). Then, a being the radius of the cylinder, the centre O of a transverse section will have a horizontal velocity aw before impact, and a velocity aw', at right angles to the radius CO, just after impact. Let c denote the distance of C from the plane on which the cylinder is rolling, R the normal and S the tangential reaction at C.

Then, for the motion of translation at right angles to CO, we have

This result shews that the cylinder will roll over the fixed point if c be less than

3

2a.

The following is a different solution of the same problem.

The motion the instant before impact is made up of two

motions, the one of translation, the velocity being aw, and the other of rotation, the angular velocity being w.

=

Let P be any point in the area of the circular section through C; let OP=r, and let the inclination of OP to CO, and its inclination to the horizontal line AB. Then the moment of the momentum of the circular section about C, due to the rotation, is equal to

The moment of the momentum, due to translation, is equal to

3

which will not be positive unless c be less than α,

that is, if

2

the cylinder be reduced to rest, c will be not less than

3

2 a.

(3) A ball, sliding without rotation along a smooth horizontal plane, impinges obliquely against a perfectly rough vertical plane: to determine the subsequent motion of the ball.

Let Ox, Oy, Oz, (fig. 250), be three rectangular axes, the plane xOy being horizontal and passing through the centre C of the ball, and the plane xOz being the rough vertical plane against which the ball impinges. Let E be the point at which the ball strikes against the vertical plane; CF the direction of

the motion of C before impact. Let u be the velocity of C before impact, a the inclination of CF to Ox; v, v,, the resolved parts of the velocity of C parallel to Ox, Oy, after the impact; w the angular velocity of the ball about C after impact; X, Y, the impulsive reactions of the rough plane along x0, and parallel to Oy, during impact; m the mass of the ball; a the radius of the ball, k the radius of gyration about a diameter.

First we will suppose the ball to be inelastic. For the motion of the ball after impact, we have

Now, the ball being perfectly inelastic, the velocity of Cat right angles to the vertical plane will be destroyed by the impact, or v1 = 0; hence, from (2),

Also, the vertical plane being perfectly rough, the ball will roll without sliding after impact: hence awv, and therefore, from (1), (3), we get

Next, let us suppose the ball to be elastic, e denoting the elasticity; then, v, v,, w', denoting on the new supposition what, w, were taken to denote on the old one, we shall have

From the equations (5) and (6), we obtain