A projectile of mass \(M\) lb., moving horizontally with a speed of \(v\) feet per second, strikes an inelastic pin of mass \(m\) lb. projecting horizontally from a block of mass \(M'\) lb., which is free to slide on a smooth plane. Prove that the pin is driven \[ x = \frac{MM'}{(M+M'+m)(M+m)}\frac{v^2}{gF} \] inches into the block, where \(F\) lb. weight is the mean resistance of the block to penetration by the pin.
A railway train is being accelerated at a certain rate when it reaches the foot of an incline. It ascends to a ridge and descends, at the same inclination, to the former level. The pull of the engine and the wheel resistance remain constant throughout. Determine the inclination of the water surface in the tank during ascent and descent, and prove that the difference between the two inclinations is twice the inclination of the slopes.
A square plate of side \(a\) and mass \(M\) is hinged about its highest edge, which is horizontal. When at rest it is struck horizontally, at a depth \(h\) below the hinge, by a particle of mass \(m\) travelling with velocity \(v\). The particle becomes embedded in the plate close to the surface. Determine the subsequent motion of the plate.
A vessel in the form of a regular tetrahedron of height \(h\) rests with one face on a horizontal table. The other faces are uniform heavy plates of weight \(w\), freely hinged about their lowest edge, and fitting closely when shut. Water is poured into the vessel through a small hole at the top, and the pressure on the sides raises the plates and opens the vessel when the height of water is \(mh\). Show that, if \(\rho w\) is the weight of water poured in, \[ 9\rho(2m^2-m^3)=2(m^2-3m+3). \]
A plane mirror is placed behind a sphere of radius \(R\) and refractive index \(\mu\). Show that the effect on a small pencil which passes directly through the sphere, is reflected at the plane, and re-traverses the sphere, will be the same as if it had been reflected at a certain mirror. Find the situation and curvature of this mirror.
The resistances of the four sides of a Wheatstone's bridge are, in order, \(\alpha, \beta, b, a\). Prove that, whatever the resistance of the bridge connecting the junctions of \(\alpha, \beta\) and \(a,b\), the resistance of the whole framework lies between \[ \frac{(a+b)(\alpha+\beta)}{a+b+\alpha+\beta} \quad \text{and} \quad \frac{a\alpha}{a+\alpha} + \frac{b\beta}{b+\beta}. \]
An insulated spherical conductor \(C\) formed of two hemispherical shells in contact (of outer and inner radii \(b,c\)) is surrounded by a concentric hollow spherical conductor \(C_1\) of internal radius \(a\), and encloses a concentric spherical conductor \(C_2\) of radius \(d\). The potential of \(C_1\) is thrice that of \(C\), while \(C_2\) is at zero potential. Find the condition that the two hemispheres of \(C\) may just be held together electrically.
The radii of the inner and outer spheres of a spherical condenser are \(a,b\). The inner sphere is excentric by a small distance \(c\). The intervening space is filled with a dielectric of specific inductive capacity \(K\). The surface density at a point on the inner sphere at an angular distance \(\theta\) from the line of centres may be represented, to the first order in \(c\), by \(\lambda_1 V(1+\lambda_2 c\cos\theta)\), where \(V\) is the potential difference. Determine \(\lambda_1\) and \(\lambda_2\).
An infinite plane has a hemispherical boss upon it, the whole forming one conductor, which is put to earth in the presence of a point charge \(e\) at a point \(P\) along the axis of the boss. The angle subtended at \(P\) by any diameter of the boss which lies in the plane is \(2\theta\). Prove that the total charge on the boss is \(e(1-\cos 2\theta/\cos\theta)\), and also that the presence of the boss increases the force of attraction on the charge at \(P\), towards the plate, in the ratio \[ (1+2\sin 2\theta \cos^2\theta\tan^2 2\theta):1. \]
The "centre of mass," \(O\), of the electricity on a conductor, charged and alone in the field, is called the electric centre of the conductor. Prove that the potential at a point \(P\) in the field must lie between \[ \frac{E}{OP}\left(1+\frac{\sigma^2}{OP^2}\right) \quad \text{and} \quad \frac{E}{OP}\left(1-\frac{\sigma^2}{2OP^2}\right), \] where \(E\) is the total charge on the conductor, and \(\sigma\) is the greatest radius of the conductor from the electric centre \(O\). Also prove that if there are two conductors \(C, C'\) in the field, with electric centres \(O, O'\), and maximum radii \(a, a'\) measured from \(O, O'\), their mutual coefficient of potential is \(1/c'\), where \(c'\) cannot differ from \(OO'\) by more than \((a^2+a'^2)/OO'\).