Forces of magnitudes \(m\text{OA}\), \(n\text{OB}\) act in the lines OA, OB respectively. Prove that the resultant force is \((m+n)\text{OC}\), where C is the point in AB such that \(m\text{AC} = n\text{CB}\). Forces \(4P, -8P, 6P, -3P\) act in the sides \(AB, BC, CD, DE\) respectively of a regular hexagon \(ABCDEF\). Find the magnitude and line of action of the resultant force.
A uniform solid rectangular block, of edges \(2a, 2b, 2c\), rests on an inclined plane, the coefficient of friction between the block and the plane being \(a/b\). The edges of length \(2a\) are parallel to the lines of greatest slope, and the edges of length \(2c\) are horizontal. A string is fastened to the block at the middle point of the highest edge of length \(2c\), and is parallel to a line of greatest slope. If the tension in the string is gradually increased, tending to pull the block up the plane, determine whether equilibrium is broken by tilting or by slipping.
Find the centre of gravity of a thin uniform hemispherical bowl. A uniform hemispherical bowl is bounded by two concentric spheres of radii \(a, b\) (\(b>a\)) and a diametral plane. Shew that the distance of the centre of gravity of the bowl from the common centre of the spheres is \(3(b^4-a^4)/8(b^3-a^3)\). A particle of mass \(m\) is fixed to the bowl at a point of the boundary common to the diametral plane and the sphere of radius \(b\), and the bowl rests in equilibrium with its curved surface on a smooth horizontal table. Prove that the axis of symmetry of the bowl makes an angle \(\theta\) with the vertical, where \[ \tan\theta = 4mb/\pi\rho(b^4-a^4), \] \(\rho\) being the density of the bowl.
Five light rods \(AB, BC, CD, DE, EF\), each of length \(2a\), are freely hinged at \(B, C, D, E\) and a light string, also of length \(2a\), joins \(A\) and \(F\). The chain so formed is wrapped round a smooth circular cylinder of radius \(a\sqrt{3}\), so that the figure assumes the form of a regular hexagon. If the string is drawn to tension \(T\), find the reactions at the hinges.
A circular wire of radius \(a\) is fixed in a vertical plane. A light elastic string of natural length \(2a\) and modulus of elasticity \((3+2\sqrt{3})W\) is fastened to the wire at its highest and lowest points. The string passes through a small smooth ring of weight \(W\) which is free to move on the wire. Shew that there is a position of equilibrium in which the radius to the ring makes an angle of \(60^\circ\) with the upward vertical, and determine its stability.
A particle slides from rest at the vertex of a smooth surface formed by revolving a parabola about its axis, the axis being vertical and the vertex upwards. Prove that the particle remains in contact with the surface. If \(v\) and \(f\) are the horizontal components of the velocity and acceleration respectively at any point, shew that the reaction at the point is proportional to \(f/v\).
A particle of mass \(m\) is fastened to one end of a light elastic string, of modulus \(mg\) and natural length \(l\), the other end of which is fixed at \(A\). The particle is released from rest at \(A\), and when it reaches its lowest point it coalesces with a particle of mass \(3m\) which is at rest. Find the distance below the fixed point at which the combined particle of mass \(4m\) comes next to rest.
A particle is projected vertically upwards with velocity \(V\) in a medium whose resistance to motion is \(kv^2\) per unit mass when the velocity is \(v\). Find the greatest height to which the particle rises. If the velocity of the particle is \(U\) when it has reached half its greatest height, shew that \[ 1 + \frac{kV^2}{g} = \left(1 + \frac{kU^2}{g}\right)^2. \]
A particle of mass \(m\) is fastened to the end \(A\) of a light rod \(AB\), and a small smooth ring is fastened to the end \(B\). The ring moves with uniform acceleration \(f\) along a horizontal wire and the rod is free to rotate about the ring in a vertical plane through the wire. Initially the rod lies along the wire and is at rest with \(AB\) in the direction of the acceleration \(f\). Find the angle which the rod makes with the vertical, when its angular velocity is next zero, and the tension in the rod at that instant.
Find the moment of inertia of a uniform elliptic lamina about a line through its centre perpendicular to its plane. If the lamina is suspended from a focus and is free to rotate in a vertical plane, shew that the length of the simple equivalent pendulum for small oscillations is \((2+3e^2)a/4e\), where \(e\) is the eccentricity, and \(2a\) the length of the major axis. A string is fastened to the lamina at the other focus, and to a fixed point at a vertical distance \(4ae\) below the fixed focus. If the tension of the string is \(T\), which may be taken to be constant, find the value of \(T\) if the length of the simple equivalent pendulum for small oscillations is halved.