equilibrium unless aw2 cos a >g: if aw3 cos a cos ẞ=g, the time of a small 2π w sin ẞ oscillation about the inclined position will be 1+tana; the 2646. A smooth semicircular disc rests with its plane vertical and vertex upwards on a smooth horizontal table and on it rest two equal uniform rods, each of which passes through two smooth fixed rings in a vertical line; the disc is slightly displaced, and in the ensuing motion one rod leaves the disc when the other is at the vertex: prove that 4 sin a − (1 + sin ẞ)* (2 – sin ß); sin2 B where m, p are the masses of the disc and of either rod, a the angle which the radius to either point of contact initially makes with the horizon, and ẞ = cos1 (2 cos a). [When the one rod leaves the disc, the pressure of the other on the disc is pg (1-sin3 ß). ] 2647. A uniform rod moves with one end on a smooth horizontal plane and the other end attached to a string which is fixed to a point above the plane; when the rod and string are in one straight line the rod is let go prove that the angular velocity of the string when vertical will be 1 h and its angular acceleration a + a, l, h being the lengths of the rod and string and the height of the fixed point above the plane respectively. 2648. A uniform beam rests with one end on a smooth horizontal table and has the other attached to a fixed point by means of a string of length : prove that the time of a small oscillation in a vertical plane will be 2649. A sphere rests on a rough horizontal plane with half its weight supported by an extensible string attached to the highest point, whose extended length is equal to the diameter of the sphere: prove that the time of small oscillations of the sphere parallel to a vertical 14a plane is 2 2650. Two equal uniform rods AB, BC, freely jointed at B, are placed on a smooth horizontal table at right angles to each other and a blow is applied to A at right angles to AB: prove that the initial velocities of A, C are in the ratio 8: 1. 2651. Two equal uniform rods AB, BC, freely jointed at B, are laid on a smooth horizontal table so as to include an angle a and a blow is applied at A at right angles to AB; determine the initial velocity of C, and prove that it will begin to move parallel to AB if 9 cos 2a = 1. 2652. Five equal uniform rods, freely jointed at their extremities, are laid in one straight line on a horizontal table and a blow applied at the centre at right angles to the line: prove that, initially, where v, v,, v, are the velocities of the three rods, w,, w, the angular velocities of the two pairs of rods, and 2a the length of each rod. 2653. Four equal uniform rods AB, BC, CD, DE, freely jointed at B, C, D, are laid on a horizontal table in the form of a square and a blow is applied at A at right angles to AB from the inside of the square: prove that the initial velocity of A is 79 times that of E. 2654. Two equal uniform rods AB, BC, freely jointed at B and moveable about A, are lying on a smooth horizontal table inclined to each other, at an angle a; a blow is applied to C at right angles to BC in a direction tending to decrease the angle ABC: prove that the initial angular velocities of AB, BC will be in the ratio cosa: 8-3 cos2 a ; that 0, the least value of the angle ABC during the motion is given by the equation 8 (5–3 cos 0) (2 – cos2 a) = (1 − cos a)2 (16 – 9 cos2 a) : also prove that, when a = π 2' the angular velocities of the rods when in a straight line will have one of the ratios 1 : 3, or 3 : - 5. 2655. A heavy uniform rod resting in stable equilibrium within a smooth ellipsoid of revolution about its major axis, which is vertical, is slightly displaced in a vertical plane: prove that the length of the equivalent simple pendulum is ace (3e2 + 1) ÷ 6 (a - c), where 2a is the length of the rod, 2c the latus rectum, and e the excentricity of the generating ellipse. 2656. A uniform rod of length 2a rests in a horizontal position with its ends on a smooth curve which is symmetrical about a vertical axis: prove that the time of a small oscillation will be r being the radius of curvature of the curve and a the angle which the normal makes with the vertical at either end of the rod. 2657. Four equal rods of length a and mass m are freely jointed so as to form a rhombus one of whose diagonals is vertical; the ends of the other diagonal are joined by an extensible string at its natural length and the system falls through a height hon to a fixed horizontal plane: prove that, if be the angle which any rod makes with the vertical at a time t after the impact, where a is the initial value of 0 and A the modulus of the string. 2658. A square is moving freely about a diagonal with angular velocity, when one of the corners not in that diagonal becomes fixed; determine the impulse on the fixed point, and prove that the instantaneous angular velocity is 4. [If V be the previous velocity of the point which becomes fixed the impulse will be MV.] 2659. A uniform heavy rod of length a, freely moveable about one end, is initially projected in a horizontal plane with angular velocity : prove that the equations of motion are where 0, are respectively the angles which the rod makes with the vertical (downwards from the fixed end) and which the projection of the rod on the horizontal plane makes with its initial position: also, if the that the resolved vertical pressure on the least value of be π 3 prove π 3 will be W, where W is the weight of the rod. 31 10 [The vertical pressure on the fixed point in any position is 2660. A uniform heavy rod moveable about one end moves in such a manner that the angle which it makes with the vertical never differs much from a prove that the time of its small oscillations will be 2661. A centre of force whose acceleration is μ (distance) is at a point 0, and from another point A at a distance a are projected simultaneously an infinite number of particles in a direction at right angles to OA and with velocities in arithmetical progression from aμ to a√μ: prove that, when after any lapse of time all the particles become suddenly rigidly connected together, the system will revolve with angular velocity 1. 1 13 [If the limits of the velocity be n‚a√μ, n ̧a √μ, and the time elapsed , the common angular velocity of the rigidly connected particles will be 3 (n, + n) √μ ÷ 6 cos2 0 + 2 (n ̧3 + n ̧n ̧ + n ̧3) sin* 0.] 2662. A uniform heavy rod is suspended by two inextensible strings of equal lengths attached to its ends and to two fixed points whose distance is equal and parallel to the length of the rod; an angular velocity about a vertical axis through its centre is suddenly communicated to the rod such that it just rises to the level of the fixed points: find the impulsive couple, and prove that the tension of either string is suddenly increased sevenfold. 2663. Two equal uniform heavy rods AB, BC, freely jointed at B, rotate uniformly about a vertical axis through A, which is fixed, with angular velocity : prove that the angles a, ẞ which the rods make with the vertical are given by the equations (8 sin a + 3 sin ẞ) cot a = (9 sin a + 6 sin ẞ) cot ß = where a is the length of each rod. 9g anzi 2664. A perfectly rough horizontal plane is made to revolve with uniform angular velocity about a vertical axis which meets the plane in 0; a heavy sphere is projected on the plane at a point P so that its centre is initially in the same state of motion as if the sphere had been placed freely on the plane at a point Q and set in motion by the impulsive friction only prove that the centre of the sphere will describe uniformly a circle of radius OQ, and whose centre R is such that OR is equal and parallel to QP. 2665. A perfectly rough plane inclined at an angle a to the horizon is made to revolve with uniform angular velocity about a normal and a heavy motionless sphere is placed upon it and set in motion by the tangential impulse: prove that the ensuing path of the centre will be a prolate, a common, or a curtate cycloid, according as the initial point of contact is without, upon, or within the circle whose equation is 202 (x2 + y2) = 35gx sin a, the axis of y being horizontal and the point where the axis of revolution meets the plane the origin. Also prove that, if the initial point of contact be the centre of this circle, the path will be a straight line. 2666. A rough hollow cylinder of revolution whose axis is vertical is made to revolve with uniform angular velocity about a fixed generator and a heavy uniform sphere is rolling on the concave surface : prove that the equation of motion is where is the angle which the common normal to the sphere and cylinder makes at a time t with the plane containing the fixed generator and the axis of the cylinder, and a + b, a are the radii of the cylinder and sphere respectively. 2667. A rough plane is made to revolve at a uniform rate about a horizontal line in itself and a sphere is set in motion upon it: determine the motion, and prove that, if when the plane is horizontal the centre of the sphere is vertically above the axis of revolution and moving parallel to it, the contact will cease when the plane has turned through an angle given by the equation 0 2668. A uniform heavy rod is free to move about one end in a vertical plane which is itself constrained to revolve about a vertical axis through the fixed end at a uniform rate N, and the greatest and least angles which the rod makes with the vertical during the motion are a, ß: prove that an2 (cos a + cos ẞ) = 3g, where a is the length of the rod : also prove that, when 3g = 2a cos a, the time of a small oscillation will be 2π 2669. Two heavy uniform rods of lengths 2a, 2b and masses A, B are freely jointed at a common end and are moveable about the other end of A, and the rods fall from a horizontal position of instantaneous rest prove that the radius of curvature of the initial path of the free end of B will be 2ab (A + B) {aA2 + b (2A + B)}. 2670. A rigid body is in motion about its centre of inertia under no forces, and at a certain instant, when the instantaneous axis is the straight line whose equations are x √ A (B − C') = ≈ √C (A − B), y = 0, a point on the cylinder A – B) (B - C) (C + A) = B (C − A) x2 (AB) + 2 (B-C) + xx is suddenly fixed: prove that the new instantaneous axis will be perpendicular to the direction of the former. (The axes of co-ordinates are, as usual, the principal axes at the centre of inertia, and A, B, C the squares of the semi-axes of the principal ellipsoid of gyration.) |