[If A describe Ox with uniform velocity U and B describe Oy at right angles to Ox, then if P be any other point fixed in the lamina and PA, PB meet the circle on AB in a, b, the acceleration of P will be always parallel to Oa and vary inversely as the cube of the distance from Ob; and, if PM be drawn parallel to Oa to meet Ob, the acceleration of P will be U. AP÷AB2. PM3.]

2430. A lamina moves in its own plane so that two points fixed in the lamina describe straight lines with equal accelerations: prove that the acceleration of the centre of instantaneous rotation is constant in direction, and that the acceleration of any point fixed in the lamina is constant in direction.

2431. Two ellipses are described about a common attractive force in their centre; the axes of the two are coincident in direction and the sum of the axes of one is equal to the difference of the axes of the other: prove that, if the describing particles be at corresponding extremities of the major axes at the same instant and be moving in opposite senses, the straight line joining them will be of constant length and of uniform angular velocity during the motion.

2432. A lamina moves in such a manner that two straight lines fixed in the lamina pass through two points fixed in space: prove that the motion of the lamina is completely represented by supposing a circle fixed in the lamina to roll with internal contact on a circle of half the radius fixed in space.

2433. A lamina moves in its own plane with uniform angular velocity so that two straight lines fixed in the lamina pass each through one of two points fixed in space: prove that the acceleration of any point fixed in the lamina is compounded of two constant accelerations, one tending to a fixed point, and the other in a direction which revolves with double the angular velocity of the lamina.

2434. A triangular lamina ABC moves so that the point A lies on a straight line be fixed in space, and the side BC passes through a point a fixed in space, and the triangles ABC, abc are equal and similar: prove that the motion of the lamina is completely represented by supposing a parabola fixed in the lamina to roll upon an equal parabola fixed in space, similar points being in contact.

2435. A particle describes a parabola under a repulsive force from the focus, varying as the distance, and another force parallel to the axis which at the vertex is three times the former; find the law of this latter force; and prove that, if two particles describe the same parabola under the action of these forces, their lines of instantaneous motion will intersect in a point which lies on a fixed confocal parabola.

[The second force is always three times the first.]

2436. Two particles describe curves under the action of central attractive forces, and the radius vector of either is always parallel and proportional to the velocity of the other: prove that the curves will be similar ellipses described about their centres.

W. P.

27

DYNAMICS OF A PARTICLE.

I. Rectilinear Motion, Kinematics.

2437. A heavy particle is attached by an extensible string to a fixed point, from which the particle is allowed to fall freely; when the particle is in its lowest position the string is of twice its natural length: prove that the modulus is four times the weight of the particle, and find the time during which the string is extended beyond its natural length.

2438. A particle at B is attached by an elastic string at its natural length to a point A and attracted by a force varying as the distance to a point C in BA produced, A dividing BC in the ratio 1 : 3, and the particle just reaches the centre of force: prove that the velocity will be greatest at a point which divides CA in the ratio 8: 7.

2439. A particle is attracted to a fixed point by a force μ (dist.)-, and repelled from the same point by a constant force f; the particle is placed at a distance a from the centre, at which point the attractive force is four times the magnitude of the repulsive, and projected directly from the centre with velocity V: prove that (1) the particle will move to infinity or not according as V> or <2af; (2) that, if x, x+c be the distances from the centre of force of two positions of the particle, the time of describing the given distance c between them will be greatest when x (x + c) = 4a. Also, when V=√2af or 3 √2af, determine the time of describing any distance.

[When V = √2af, the time of reaching a distance x from the centre

of force is

2 (

Jj Zx + J
{√za-√2a - Jalog (√ √2) + 2 Ja log (√/2 + 1)};

and, when V3/2af, the time is

=

2440. The accelerations of a point describing a curve are resolved into two, along the radius vector and parallel to the prime radius: prove that these accelerations are respectively

2441. The motion of a point is referred to two axes Ox, Oy, of which Ox is fixed and Oy revolves about the origin: prove that the accelerations in these directions at any time t are

2442. A point P is taken on the tangent to a given curve at a point Q, and O is a fixed point on the curve, the arc OQ=8, QP=r, and is the angle through which the tangent revolves as the point of contact passes from 0 to Q: prove that the accelerations of P in direction QP and in the direction at right angles to this, in the sense in which increases, are respectively

2443. A point describes a curve of double curvature, and its polar co-ordinates at the time t are (r, 0, p): prove that its accelerations (1) along the radius vector, (2) perpendicular to the radius vector in the plane of and in the sense in which increases, and (3) perpendicular to the plane of in the sense in which increases, are respectively

2444. A point describes a parabola in such a manner that its velocity, at a distance r from the focus, is (c), where ƒ, c are constant: prove that its acceleration is compounded of ƒ parallel to the axis and along the radius vector from the focus.

зов

2445. A point describes a semi-ellipse bounded by the minor axis, and its velocity at a distancer from the focus is a

'ƒ (a− r)
r (2a-r)'

where 2a is the length of the major axis and ƒ a constant acceleration: prove that the acceleration of the point is compounded of two, each varying inversely as the square of the distance, one tending to the nearer focus and the other from the farther focus.

2446. A point is describing a circle, and its velocity at an angular distance ✪ from a fixed point on the circle varies as √1+ cos2 0 ÷ sin2 0 : prove that its acceleration is compounded of two tending to fixed points at the extremities of a diameter, each varying inversely as the fifth power of the distance and equal at equal distances.

2447. A point describes a circle under acceleration, constant, but not tending to the centre: prove that the point oscillates through a quadrant and that the line of action of the acceleration always touches a certain epicycloid.

2a 3

[The radius of the fixed circle of the epicycloid is and of the moving circle

α

6

a being the radius of the circle described by the point.]

2448. A parabola is described with accelerations F, A, tending to the focus and parallel to the axis respectively: prove that

2449. A point describes an ellipse under accelerations F1, F, tending to the foci, and r1, r, are the focal distances of the point: prove

that

2

2450. The parabola y = 4ax is described under accelerations X, Y parallel to the axes: prove that

2451. A point describes a parabola under acceleration which makes a constant angle a with the normal, and is the angle described from the vertex about the focus in a time t: prove that

0 tan a

∞ ( 1 + cos 0);

dt

and find the law of acceleration.

0
2

[The acceleration varies as cos-etana, which is easily expressed as a function of the focal distance.]

2452. A point P describes a circle of radius 4a with uniform angular velocity w about the centre, and another point Q describes a circle of radius a with angular velocity 2w about P: prove that the acceleration of Q varies as the distance of P from a certain fixed point.

2453. The only curve which can be described under constant acceleration in a direction making a constant angle with the normal is an equiangular spiral.

2454. An equiangular spiral is described by a point with constant acceleration in a direction making an angle & with the normal: prove that

a being the constant angle of the spiral and the angle through which the radius vector has turned from a given position.

2455. The parabola y = 2cx is described by a point under acceleration making a constant angle a with the axis and the velocity when the acceleration is normal is V: prove that, at any point (x, y) of the parabola, the acceleration is V (c cos ay sin a); and that, if when the acceleration is normal the particle is moving towards the vertex, the time in which the direction of motion will turn through a right angle will be c÷V sin 2a cos a.

2456. A lamina moves so that two straight lines fixed in it pass through two points fixed in space and the angular velocity is uniform: prove that any point fixed in the lamina, whose distance from the point of intersection of the two straight lines is twice the diameter of the circle described by that point, will move under acceleration whose line of action always touches a three-cusped hypocycloid,

2457. The catenary 8 = c tan is described under acceleration which at any point makes an angle & with the normal on the side towards the vertex: prove that the acceleration varies inversely as the cube of the distance from the directrix.

2458. A point describes a parabola, starting from rest at the vertex, under acceleration which makes with the tangent an angle tan' (2 tan 0), where is the angle through which the tangent has turned: prove that the acceleration varies as 4r-3a. r-2, where r is the focal distance and a the initial value of r.

2459. The curve whose intrinsic equation is sa tan 26 is described by a point under constant acceleration: prove that the direction of the acceleration makes with the tangent an angle - 24, where O is given by the equation

2460. A point describes an epicycloid under acceleration tending to the centre of the fixed circle: prove that the pedal of the epicycloid with respect to the centre will also be described under acceleration tending to the same point.