A sphere of radius \(a\) has centre \(O\), and \(P\) is a point distant \(z\) from \(O\). Find the mean value with respect to area of the \(n\)th power of the distance of the surface of the sphere from \(P\), where \(n \ge -1\) and is not necessarily integral, distinguishing between the cases when \(z > a\) and \(z < a\). Verify that if \(P\) is external to the sphere and \(P'\) is the inverse of \(P\) with respect to the sphere, the mean value for \(P'\) is \((\frac{a}{z})^n\) that for \(P\).
A tripod consists of three uniform rods \(AB, AC\) and \(AD\), each of length \(l\) and weight \(W\), smoothly jointed at \(A\). It rests in the form of a regular tetrahedron, with apex \(A\), upon a smooth horizontal surface. The feet \(B\) and \(C\) are fixed (the rods \(AB\) and \(AC\) being free to rotate about these points), and the foot \(D\) is prevented from slipping by inextensible strings \(BD\) and \(CD\). A horizontal force \(F\), in the direction of the perpendicular from \(D\) to \(BC\), acts at \(A\). Calculate (i) the force of interaction between \(AD\) and the surface, (ii) the tension in the strings. If the magnitude of the applied force is gradually increased, for what value of \(F\) will equilibrium be broken?
\(ABC\) is a plane triangular lamina. The sides \(BC, CA, AB\) are divided internally and externally in the ratio \(2:1\) by the points \(D\) and \(D'\), \(E\) and \(E'\), \(F\) and \(F'\) respectively. Six forces whose magnitudes and lines of action are represented by \(k\vec{AD}, k\vec{BE}, k\vec{CF}, k'\vec{D'A}, k'\vec{E'B}, k'\vec{F'C}\) are applied to the lamina. Show that, for a certain value of \(k'/k\), the forces will be in equilibrium.
The fixed rods \(OX\) and \(OY\) lie in a vertical plane and are each inclined to the upward vertical at an acute angle \(\alpha\). The ends \(A, B\) of a light rod of length \(l\) can slide along \(OX, OY\) respectively, the angle of friction at \(A\) and at \(B\) being \(\lambda\). The rod is initially horizontal, and a downward vertical force is then applied to it at a point at a distance \(k\) from the mid-point of the rod. Show that (i) if \(\lambda < \alpha\) and \(\lambda+\alpha < \frac{1}{2}\pi\) equilibrium will be broken if \(k\) exceeds a certain value (to be found), and (ii) if \(\lambda > \alpha\) or \(\lambda+\alpha > \frac{1}{2}\pi\) equilibrium cannot be broken whatever the value of \(k\).
A uniform plank of thickness \(t\) is placed symmetrically across a rough fixed horizontal log whose cross-section is a circle of radius \(a\), and rests in equilibrium in a horizontal position. Discuss its stability for all values of \(t/a\).
(i) When a rod of natural length \(l\) cm. and cross-section \(S\) cm.\(^2\) is under tension \(T\) dynes, its extension of length is \[ \frac{Tl}{ES} \text{ cm.}, \] where \(E\) is Young's Modulus. Write down the relation between the values \(E\) and \(E'\) of Young's Modulus in the C.G.S. and M.K.S. (Metre, Kilogram, Second) systems of units, respectively. (ii) When liquid flows through a long straight tube of length \(l\) and circular cross-section of radius \(a\), the volume flowing per unit time is \[ \frac{\pi a^4 \Delta p}{8\eta l}, \] where \(\Delta p\) is the difference of pressure at the two ends and \(\eta\) is a certain constant characteristic of the liquid. When a sphere of radius \(r\) moves slowly through a large volume of liquid with steady velocity \(U\), the resisting force \(F\) depends only on \(r, U\) and \(\eta\). Find how \(F\) depends on \(r, U\) and \(\eta\).
A wedge of mass \(M\) is placed upon a horizontal table; the sloping face makes an angle \(\alpha\) with the table. A particle of mass \(m\) is placed upon the sloping face at a point at a height \(h\) above the table. The system is then released from rest. Assuming that the wedge slides without rotation and that friction is everywhere negligible, find the force of reaction between the particle and the wedge. Show that the particle reaches the table in time \[ \left[ \frac{2h (M+m\sin^2\alpha)}{(M+m)g\sin^2\alpha} \right]^{\frac{1}{2}}. \]
A particle of mass \(m\) is projected vertically upwards with speed \(v_0\). The resistance to its motion when its speed is \(v\) is \(kmv^2\). Show that it reaches the height \[ \frac{1}{2k} \log\left(1+\frac{kv_0^2}{g}\right). \] If it returns to the point of projection with speed \(v_1\), prove that \[ \frac{1}{v_1^2} - \frac{1}{v_0^2} = \frac{k}{g}. \]
One end \(A\) of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda\) is fixed. The other end is attached to a particle of mass \(m\) which moves on a smooth horizontal table at a depth \(b\) below \(A\). If the particle moves in a circle with constant angular velocity \(\omega\) and with the string inclined at a constant angle \(\alpha\) to the vertical, prove that \[ b\omega^2 \le g, \quad mab\omega^2 = \lambda (b - a\cos\alpha). \] Deduce that \(\omega\) must satisfy the conditions \[ \lambda(b-a) < mab\omega^2 < \lambda b \] and that no such motion (whatever the values of \(\omega\) and \(\alpha\)) is possible if the particle can hang in equilibrium without reaching the table.
Two spheres, of masses \(m_1\) and \(m_2\), move without rotation along the same straight line with velocities \(u_1\) and \(u_2\). Their centre of mass is \(G\). Prove that their total kinetic energy is equal to the sum of (a) the kinetic energy of a particle of mass \(m_1+m_2\) moving with the velocity of \(G\), (b) the kinetic energies of the two spheres moving with velocities equal to their actual velocities relative to \(G\). Prove that, if the spheres impinge directly, the first term (a) is unaltered and the second term (b) is reduced to a fraction \(e^2\) of its original value, where \(e\) is the coefficient of restitution.