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.
A given set of coplanar forces reduces to a single resultant force, and is such that the total moment about a point \(O\) is \(Q\), while the sums of the components parallel to two perpendicular lines \(Ox, Oy\) are \(X\) and \(Y\) respectively. Find the equation of the line of action of the resultant. Forces equal to 1, 4, 2, and 6 lb. weight act along the sides \(OB, BC, CD, DO\) respectively of a square \(OBCD\) with side of length \(a\). Find the magnitude of their resultant and obtain the equation of the line of action referred to \(OB\) and \(OD\) as coordinate axes.
Obtain an expression for the ratio of the tensions at the two ends of a rope wound round a post of uniform coefficient of friction when the rope is in limiting equilibrium. A sailor is holding a ship by means of a horizontal rope wound round a post of a wharf. The coefficient of friction is 1/3. Find the maximum force that can be exerted by the ship if the sailor is to exert a pull of not more than 100 lb. and the rope is wrapped \(2\frac{1}{2}\) times round the post.
A uniform solid elliptic cylinder is in stable equilibrium resting on a perfectly rough horizontal table. Show that no amount of loading along its highest generator will render it unstable if the eccentricity of the orthogonal cross-section exceeds a certain value, to be found.