A long thin uniform plank of weight \(W\) lies symmetrically along the corner at the bottom of a smooth wall. Its breadth makes an angle \(\alpha\) with the ground. A horizontal force \(F\) acting along the line of contact of the plank with the ground is applied to one end of the plank. If \(\mu\) is the coefficient of friction between the plank and the ground, find the least value of \(F\) that will cause the plank to move.
An ancient catapult consists of a uniform lever arm \(ABC\) of mass \(3M\) through \(\frac{1}{4}\pi\) and pivoted at \(B\). \(AB\) is above \(BC\) which is horizontal and of length \(l\). The projectile of mass \(m\) fills the cupped end at \(A\) which is a heavy weight of mass \(M\) is banked to a height and dropped onto \(C\). Assume that the weight remains in contact with \(C\) and that the projectile leaves as soon as it feels the impulse. If \(AB\) is of length \(\frac{l}{4}\), show that the value $$x = \left(\frac{M + M'}{m + M'}\right)^{\frac{1}{4}}$$ will give the greatest range. What is that range?
A flexible trans-Atlantic cable of density \(\rho\) and radius \(r\) hangs over a cliff in the ocean floor which forms part of the edge of the continental shelf. The density of sea water is \(\rho'\), the height of the cliff \(h\) and the length of cable hanging free between the cliff-top and where the cable again touches the horizontal surface of the mud is \(s\). Find the greatest tension in the cable.
A normal bicycle is constrained to remain in a vertical plane. Its wheels are rough. The lower of the pedals is pushed horizontally towards the back wheel by a person \(P\), who describes the motion that ensues and the sense of rotation of the back wheel. Does it matter whether \(P\) is squatting beside the bicycle or sitting on it as he pushes the pedal? Explain the forces and couples that act in and on the bicycle to cause the motion. [In any diagram the front wheel of the bicycle should be on the left.]
A system of particles of masses \(m_i\) are at positions \(x_i\) on a line and are subject to known forces \(F_i\) in the direction of increasing \(x\). These \(F_i\) are of the form $$F_i = K(t)m_iv_i,$$ where \(K\) is a known function of time \(t\) which is the same for all particles. If the system starts from rest at \(t = 0\) show that the motion of all the particles can be described by multiplying the initial coordinate of each particle by a 'scaling factor' which depends on time but not on the particle. If \(K(t) = n(n-1)(t+t_0)^{-2}\), where \(n > 1\), \(t_0 > 0\), find the scaling factor as a function of time.
A bead is released from rest on a rigid smooth wire in the shape of cycloid arc with its cusps pointing vertically up. Show that it oscillates with a period independent of its initial position. If the particle is released from a cusp, show that the reaction on the wire is always twice as great as if the bead were sliding on a straight wire, with the same slope as the point where it is on the cycloid. (The intrinsic equation of a cycloid is \(s = 4a\sin\psi\).)
A right circular cylinder of radius \(a\) and radius of gyration \(k\) is projected with velocity \(V\) and zero spin up a plane inclined at an angle \(\alpha\) to the horizontal. The coefficient of friction between them is \(\mu\). Show that the cylinder stops slipping after a time $$\frac{k^2V}{g[\mu(a^2 + k^2)\cos\alpha + k^2\sin\alpha]}.$$ Find the condition that it starts to roll, and whether it continues to do so when this condition is satisfied.
A gun (with fixed muzzle velocity) is on a plane inclined at an angle \(\alpha\) to the horizontal. It can fire in any direction and at any elevation. Show that the part of the plane which can be hit is the interior of an ellipse of eccentricity \(\sin\alpha\), with one focus at the position of the gun.
Two similar simple pendulums of length \(l\) are suspended at the same height. They have light bobs attached to the opposite ends of a light inextensible string also of length \(l\), so that they are pulled together, and the pendulums make a small angle \(s\) with the vertical. The pendulums are displaced at right angles to the original plane of the system through angles small compared with \(s\). Assuming that all the tensions maintain their original values to the degree of approximation necessary, show that the subsequent displacements of the pendulums can be represented by the sum and difference of two harmonic oscillations with slightly different frequencies. Deduce that if only one pendulum is displaced, the motion is concentrated in the other after a time approximately \((n/\alpha)\sqrt{(l/g)}\). Describe the motion.
A rigid lamina bounded by a simple closed curve is rolling along a straight line in its plane. Find the condition that the rate of change of the moment of momentum of the lamina about the point of contact with the line should be equal to the moment of the forces on the lamina about that point. Show that these quantities remain equal while the lamina rolls through a complete revolution if, and only if, the lamina is circular and its centre of mass is at the centre of the circle.