The centre of a circular disc of radius \(r\) is \(O\), and \(P\) is a point on the line through \(O\) perpendicular to the plane of the disc such that \(OP=p\). Prove that the mean distance with respect to area of points of the disc from \(P\) is \[ \frac{2}{3r^2}\{(p^2+r^2)^{3/2}-p^3\}. \] Find the mean distance with respect to volume of the interior points of a sphere of radius \(a\) from a fixed external point at distance \(c\) from its centre.
Two ladders, \(AB, BC\), each of weight \(w\) and length \(2a\), and with their centres of gravity at the middle points, are freely hinged together at \(B\), and rest with their lower ends \(A, C\), on a smooth horizontal floor. A string of length \(2l \ (<2a)\) joins the middle points of the ladders, and carries at its mid-point a weight \(2W\). If the system is in equilibrium with the angle \(ABC\) equal to \(2\theta\), show that \(\theta\) is either zero or is given by \[ (w^2+2Ww) a^2 \cos^2\theta = (w+W)^2(a^2-l^2). \]
Nine equal light straight rods \(AB, BC, CD, DE, EF, AC, CE, BD, DF\) are freely jointed together, to form a plane framework. This framework is freely hinged to a wall at \(A\), so that \(BDF\) and \(ACE\) are horizontal, with \(BDF\) above \(ACE\) and in the same vertical plane. A light chain, making an angle of \(30^\circ\) with the vertical, joins \(F\) to a point of the wall vertically above \(A\). A load of \(900\) lb.-wt. hangs from \(E\). Find graphically or by calculation the tension in the chain, and the magnitude and direction of the force exerted at \(A\) by the wall on the framework. Find the forces in the rods \(CE\) and \(DF\), stating in each case whether the rod is in tension or in compression.
Prove that the mass centre of a uniform solid hemisphere of radius \(a\) is situated at a distance \(\frac{3}{8}a\) from the plane face. From a uniform solid cube, of edge \(2a\) and density \(\rho\), a hemispherical portion of radius \(a\) is removed, the centre of which coincides with the centre of one of the faces of the cube. The cavity is filled with solid material of density \(2\rho\), so that the external form of the solid is a cube. Find the distance of the mass centre of the composite solid from the centre of the cube.
A uniform square lamina \(ABCD\), of weight \(W\), rests in a vertical plane under the action of a force \(P\), as shown in the figure. The lamina is in contact with a smooth horizontal floor and a rough vertical wall, at which the coefficient of friction is \(\mu\). The plane of the lamina is perpendicular to the wall, and its uppermost edge \(AB\) makes an angle \(\theta (<\frac{1}{4}\pi)\) with the horizontal. The force \(P\) at \(A\) acts horizontally in the plane of the lamina towards the wall. Find the horizontal and vertical components of the reactions on the lamina at \(C\) and \(D\). Prove that equilibrium is impossible if \(\mu < 2\tan(\frac{1}{4}\pi - \theta)\). % Diagram shows a square lamina ABCD. A is top-left, B top-right, C bottom-right, D bottom-left. % D and C are on a horizontal floor. The right edge BC is against a vertical wall. % The edge AB makes an angle theta with the horizontal. % A horizontal force P acts at A, pushing towards the wall.
To a man travelling at 10 m.p.h. due eastwards over level country the wind appears to blow from the north-east, while the wind appears to a man travelling due westwards at 30 m.p.h. to blow from the north-west. Find the velocity and direction of the wind, supposed steady. Find the direction or directions in which the man would have to travel at 15 m.p.h. in order that the wind should appear to blow from due north.
State Newton's Laws of Motion. A force of magnitude \(T\) lb.-wt., acting vertically upwards, is applied to a mass of 60 lb., which is at rest at the instant \(t=0\). The values of \(T\) at times \(t\) (sec.) are:
A particle is projected from a point on level ground with velocity \(V\). Show that, if the effect of air resistance is neglected, the maximum range of the particle is \(V^2/g\). Water is pumped from a sump in which the free surface is 20 ft. below ground level. It is delivered to a nozzle, of cross-sectional area 8 sq. in., which is situated on the level ground. At the nozzle the speed is such that the jet can just reach a point on the ground 200 ft. away. Taking the density of water to be 62.5 lb. per cu. ft., and neglecting frictional losses, calculate the horse-power of the pump.
A trolley, of mass \(M\), can roll without friction on rails on a horizontal table. A light string is fastened at one end to the trolley and passes through a fixed smooth ring, so that the upper part of the string is horizontal and in the direction of the rails, while the other end of the string carries a particle of mass \(m_1\). The moments of inertia of the trolley wheels are negligible. A particle, of mass \(m_2\), lies on the upper surface of the trolley, which is horizontal and rough, with coefficient of friction \(\mu\). Show that motion is possible in which \(m_1\) moves vertically, with the string taut and with \(m_2\) continuing at rest relative to the trolley, provided that \(m_1 < \mu(M+m_1+m_2)\). Supposing this condition satisfied, show that, if \(m_2\) be projected horizontally along the trolley with velocity \(v_0\) parallel to the upper part of the string and away from the fixed ring at an instant when \(m_1\) and the trolley are at rest, then slipping between \(m_2\) and the trolley will cease after a time \[ \frac{(M+m_1)v_0}{\{\mu(m_1+m_2+M) - m_1\}g}. \]
If a point \(P\) is moving in a circle of radius \(r\) with constant angular velocity \(\omega\) about the centre \(O\), prove that the acceleration of \(P\) is directed towards \(O\) and that its magnitude is \(r\omega^2\). A vertical cylindrical shaft is constrained to rotate with constant angular velocity \(\Omega\) about its axis, and a light horizontal rod is freely hinged to the surface of the shaft so that the hinge-line is vertical. The other end of the rod carries a small block of mass \(M\), which bears against the rough inner surface of a fixed concentric tube, of internal radius \(b\) and with coefficient of friction \(\mu\). If the rod makes an angle \(\theta\) with the radius through \(M\), show that the force exerted on the tube has a moment about the axis equal to \[ \mu M b^2 \Omega^2 \sin\theta/(\sin\theta + \mu\cos\theta). \] Find an expression for the rate at which energy is being dissipated by friction.