A particle of mass 2 lb. is placed on the smooth face of an inclined plane of mass 7 lb. and slope \(30^\circ\), which is free to slide on a smooth horizontal plane in a direction perpendicular to its edge. Shew that if the system start from rest the particle will slide down a distance of 15 feet along the face of the plane in 1.25 seconds.
Find the values of \(\sin 15^\circ\) and \(\sin 18^\circ\). If \[ \cos(\theta-\phi)/\cos(\theta+\phi) = a/b, \] prove that \[ (a^2+b^2-2ab\cos 2\theta)(a^2+b^2-2ab\cos 2\phi) = (a^2-b^2)^2. \]
In any triangle prove that \[ r=4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}, \quad s=4R\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}. \] A perpendicular \(CD\) is drawn from the vertex \(C\) of a right-angled triangle on the hypotenuse \(AB\). Prove that the square of the radius of the circle inscribed in \(ABC\) is equal to the sum of the squares of the radii of those inscribed in \(ACD\) and \(BCD\).
Expand \(\cos n\theta\) in a series of ascending powers of \(\cos\theta\). Prove that \[ \sum_{r=0}^{r=n-1} \sec^2\left(\theta+\frac{2r\pi}{n}\right) = n^2\sec^2 n\theta, \] when \(n\) is odd, and find its value when \(n\) is even.
Prove that \((\cos\theta+i\sin\theta)^{p/q}\) has \(q\) values, where \(p,q\) are integers and \(q\) is prime to \(p\). If \[ \tan^{-1}(\xi+i\eta) = \sin^{-1}(x+iy), \] prove that \[ \xi^2+\eta^2 = (x^2+y^2)/\sqrt{x^4+y^4+2x^2y^2-2x^2+2y^2+1}. \]
Four uniform rods freely jointed form a parallelogram \(ABCD\), the weights of the opposite sides being equal. A weightless rod connects \(B\) and \(D\), and the system is suspended freely from \(A\). Prove that the stress in the rod \(BD\) is \(\frac{1}{2}W \cdot BD/AC\), where \(W\) is half the weight of the system.
State the laws of limiting friction. A uniform rod \(AB\) of weight \(W\) rests with one end \(A\) on a rough horizontal plane and the other end against an equally rough vertical wall, the vertical plane through the rod being perpendicular to the wall. Show that for equilibrium the inclination \(\alpha\) of the rod to the horizontal must be greater than \(\pi/2-2\epsilon\), where \(\epsilon\) is the angle of friction. If \(\alpha\) is less than \(\pi/2-2\epsilon\), show that the least force acting on \(A\) which will just maintain equilibrium is \[ W \cos(\alpha+2\epsilon)/2\sin(\alpha+\epsilon), \] and find the direction of this force.
A string passing over a smooth pulley carries a mass \(4m\) at one end and a pulley of mass \(m\) at the other. A string carrying masses \(m\) and \(2m\) at its ends passes over the latter pulley. Find the acceleration of the mass \(4m\) when the system is moving freely under gravity.
A particle is projected with a given velocity \(v\) from the foot of an inclined plane of slope \(\alpha\). The direction of projection lies in a plane containing the line of greatest slope and makes an angle \(\theta\) with the face of the plane. Prove that if the particle strikes the plane perpendicularly \(\cot\theta = 2\tan\alpha\). Show that, for different values of \(\alpha\), the range on the plane when the particle strikes it perpendicularly cannot be greater than \(v^2/g\sqrt{3}\).
Two smooth rings \(A, B\), each of mass \(m\), can slide on a smooth horizontal wire; a light string \(ACB\) has its ends attached to the rings and has an equal mass \(m\) attached to it at its middle point \(C\). The rings \(A, B\) are released from rest when the angle \(ACB\) is \(60^\circ\). Prove that the mass \(C\) begins to descend with an acceleration \(g/7\).