when there is a position of equilibrium other than the highest or lowest point.

210. Find the form of the tube in order that the particle projected with given velocity may preserve its velocity unchanged, gravity acting parallel to the axis.

Resolving tangentially, and taking co-ordinates x, y in the plane of the curve, the axis of revolution being that of y, we have

dx dy -9 ds'

2g

Hence, 2=2(y+k), the equation to a parabola whose axis is vertical and vertex downwards. This result might easily have been foreseen, as the velocity can only be constant if the accelerating effect of the impressed forces along the curve be zero at every point; that is, if the resultant of gravity and centrifugal force lie in the normal. That this may be the case, we must have Centrifugal force: Gravity :: Ordinate: Subnormal. But the centrifugal force is proportional to the ordinate, hence the subnormal must be proportional to gravity, i. e. must be constant: a property peculiar to the parabola. This proposition has a singular application in Hydrostatics.

211. A particle moves on a rough curve, under the action of given forces; to determine the motion.

If ' be the coefficient of kinetic friction, and

R = √(R122 + R22)

2

be the force of constraint as in § (186), the effect of friction will be a force μ√(R+R) acting in the tangent to the curve, and in the opposite direction to the particle's motion.

Equation (1) of § (186), will therefore become

d2s

dt

= S− μ' √ (R ̧2 + R ̧2),

the other two equations remaining the same.

If from the three we eliminate R, and R2, we may by means of the equations to the curve eliminate x, y and z, and the final result, involving only s and t, suffices to determine the motion completely.

212. Ex. A particle moves in a rough tube in the form of a plane curve, under the action of no forces; to determine

the motion.

=

Hence, vae, where a is the velocity when 0.

It may be instructive to compare this result with that for the tension of a string stretched over a rough curve.

ds

If the curve be one of double curvature, is the angle

ρ

between two successive tangents. If the surface of which the curve is the cuspidal edge be developed, and if represent the angle between the tangents corresponding to the initial and final positions of the particle,

213. A particle under the action of given forces moves on a given rough surface; to determine the motion.

If R be the force of constraint due to the surface, the effect of friction is 'R acting in the tangent to the path of the particle, and the equations of § 191 become

from which R must be eliminated. The two resulting equations contain x, y, z and t, and if the latter be eliminated, we have one equation in x, y, z which, with the equation to the surface, will completely determine the path. In general these equations are utterly intractable.

EXAMPLES.

(1) If a particle attached by a string to a point just make complete revolutions in a vertical plane, the tension of the string in the two positions when it is vertical is zero, and six times the weight of the particle, respectively.

(2) On a railway where the friction is of the load,

1 240

shew that five times as much can be carried on the level as up an incline of 1 in 60 by the same power at the same rate.

(3) A pendulum which vibrates seconds at a place A, gains two beats per hour at a place B; compare the weights of any the same substance at the two places.

(4) From a point upon the surface of a smooth vertical circular hollow cylinder, and inside, a particle is projected in a direction making an angle a with the generating line through the point; find the velocity of projection that the

particle may rise to a given height (h) above the point, and the condition that the highest point may be vertically above the point of projection.

(5) A heavy particle rests on the arc of a smooth vertical circle at an angular distance of 30° from the lowest point, being repelled from one extremity of the horizontal diameter by a constant force; shew that, if slightly displaced along the arc, it will perform small oscillations in the time

(6) A particle is constrained to move on a smooth curve under the action of a central force P tending to the pole, and the pressure on the curve varies always as the curvature, shew that

(7) A seconds pendulum when taken to the top of a mountain h miles high will lose 21.6h beats in a day nearly.

(8) AB is the diameter of a sphere of radius a; a centre of force at A attracts with a force (ux distance); from the extremity of a diameter perpendicular to AB a particle is projected along the inner surface with a velocity (2) a: shew that the velocity of the particle at any point P is proportional . to sin 0, and the pressure to 1-3 sin' 0, where is the angle PAB.

(9) A chord AB of a circle is vertical and subtends at the centre an angle 2 cotμ. Shew that the time down any chord AC drawn in the smaller of the two segments into which AB divides the circle is constant, AC being rough and the coefficient of friction.

μ

(10) A particle under the action of no force is projected with velocity V in a rough tube in the form of an equiangular spiral at a distance a from the pole and towards the pole; shew that it will arrive at the pole in time

μ

a being the angle of the spiral and μ (> cot a) the coefficient of friction.

(11) If a particle move on the surface of a smooth cone with its axis vertical and vertex downwards, and gravity be the only force acting, shew that the differential equation of the projection of its path on a horizontal plane is

a being the semi-vertical angle of the cone.

(12) A particle is suspended from a fixed point by an inextensible string: find the velocity with which it must be projected when at the lowest point, so that its path after the string has ceased to be stretched may pass through the point of suspension.

(13) A particle is constrained to remain on the curve

. — r = a (1 − cos 0) and is repelled from the pole by a force

if its velocity at the apse be equal to (")

μ

==

μ shew that it will

a

3

arrive at the initial line again in time π

a

(14) A particle slides down a catenary, whose plane is vertical and vertex upwards, the velocity at any point being that due to falling from the directrix; prove that the pressure at any point is inversely proportional to the distance of that point from the directrix."

(15) A particle projected with given velocity, moves under the action of gravity on a curve in a vertical plane; find the nature of the curve that the pressure on it may be the same throughout the motion.

(16) A particle is projected with given velocity from the