If three forces acting on a body are in equilibrium, show that they are coplanar and either concurrent or parallel. A smooth sphere of radius \(a\) and weight \(w\) is suspended by a string of length \(l\) from a hook. From the same hook a weight \(w'\) is hung by a long string. Shew that the angle which the first string makes with the vertical is \(\sin^{-1} \frac{w'a}{(w+w')(a+l)}\).
From the points B, C, D of a light string ABCDE weights proportional to 4, 8 and 5 are hung respectively. It is found that the portions of the string BC and CD make angles of 25\(^\circ\) and 15\(^\circ\) respectively with the horizontal. Find graphically the angles which the strings AB and DE make with the vertical.
A triangular frame formed of three uniform rods, jointed together at their extremities, of length 3, 4 and 5 ft respectively, is suspended by a string attached to the middle point of the longest side. Shew that the reactions at the joints are \[ W\sqrt{\frac{137}{8}}, \quad W\sqrt{\frac{109}{8}}, \quad W\frac{5}{2\sqrt{2}}, \] where \(W\) is the weight per foot of rod.
Out of a circular disc of metal a circle is punched whose diameter is a radius \(OA\) of the disc. The disc is then placed vertically resting on two rough parallel rails in the same horizontal plane, the plane of the disc being perpendicular to the rails. The chord of contact subtends an angle \(2\alpha\) at the centre of the disc. Shew that if the angle which \(OA\) makes with the vertical is greater than \(\sin^{-1}\left(\frac{3\sin 2\epsilon}{\cos\alpha}\right)\) where \(\epsilon\) is the angle of friction, the disc will slip.
In rectilinear motion, when the acceleration at consecutive intervals of time is given, shew how the velocity can be calculated by plotting the acceleration-time curve. A motor car is running at 10 M.P.H. when it starts to accelerate. The acceleration diminishes uniformly with the time and after 20 seconds the acceleration is zero and the car is running at 25 M.P.H. Sketch the velocity-time curve and calculate the distance travelled during the period of acceleration and the time that elapses before the speed of 20 M.P.H. is reached.
Two weights \(A\) and \(B\) are connected by a string passing over a smooth light pulley. To the weight \(B\) is attached another weight \(C\) by a string of length 2 ft. \(B\) and \(C\) are held initially in contact and resting on a platform vertically below the pulley. If the masses of \(A, B\) and \(C\) are 5, 3 and 4 lb. respectively, shew that when the system is free to move, the weight \(C\) will strike the platform again after \(12/\sqrt{g}\) seconds and that the weight \(B\) will come momentarily to rest at a distance \(1\frac{1}{2}\) ft from the platform.
A train of mass 200 tons is ascending an incline of 1 in 100, the resistance to the motion being 15 lb. wt per ton. When its velocity has reached 12 M.P.H., what is its acceleration if the H.P. then developed by the engine is 600?
A pile of mass 4 tons is to be driven into the muddy bottom of a canal, the resistance varying directly as the depth already penetrated. It is found that the pile sinks a distance of 3 inches under its own weight. Shew that when the pile-hammer of mass \(\frac{1}{2}\) ton falls on the pile through a distance of 12 ft, the pile and the hammer sink a further distance of 1.47 inches.
Prove that the path of a projectile in a vacuum would be a parabola. A small elastic spherical ball is dropped on to a fixed hemispherical dome, whose base rests on a horizontal plane, striking it at a point where the radius from the centre of the dome to the point of contact makes an angle 30\(^\circ\) with the vertical. Shew that the ball will strike the plane at a distance \(R\) from the dome if it is dropped from a height \(\frac{R}{4}(9-3\sqrt{3})\) above the horizontal plane, where \(R\) is the radius of the dome and the coefficient of elasticity is \(\frac{1}{\sqrt{3}}\).
A particle of mass \(m\), lying on a smooth horizontal table, is attached to two elastic strings whose natural lengths are \(l\) and \(l'\) and moduli \(\lambda\) and \(\lambda'\) respectively. The other ends of the strings are fixed to two points on the table at a distance apart greater than \(l+l'\). Shew that if the particle vibrates in the line of the strings, its period will be \[ 2\pi \sqrt{\frac{m}{\frac{\lambda}{l}+\frac{\lambda'}{l'}}}. \]