A wooden body of mass \(5m\) is projected at an angle to the vertical from a point of a horizontal plane. When it is at the highest point of its trajectory it is hit by a bullet of mass \(m\) flying vertically upwards. The bullet becomes embedded in the body which then falls on the plane at the same point as if the bullet had not hit it. Show that if the bullet was fired from a point of the plane its muzzle velocity must have been \((2\cdot21)^{\frac{1}{2}}\) times the vertical component of the velocity of projection of the wooden body.
A small ring of mass \(m\) can slide on a fixed smooth wire which is in the form of a single arc of the cycloid, \(x=a(\theta-\sin\theta)\), \(y=-a(1-\cos\theta)\), from \(\theta=0\) to \(\theta=2\pi\), the positive \(y\)-axis being vertically upwards. The ring is released from rest at the point \(x=0, y=0\). Prove that its a vertical velocity is greatest when it has fallen through a vertical distance equal to \(a\). Calculate the time taken by the ring in falling to the lowest point of the wire.
The power output of a car at speed \(v\) is \[ W \frac{v^3 w^2}{(v^2+w^2)^2}, \] where \(W\) is the weight of the car and \(w\) is 30 m.p.h., so that, if the weight of the car is \(1\frac{1}{4}\) tons, its power at 30 m.p.h. is 56 h.p. The car climbs a hill inclined at an angle \(\alpha\) to the horizontal. Show that if \(\sin\alpha > \frac{3}{32}\) its speed will decrease on the hill in all circumstances. Find the inclination of the steepest hill on which the car can maintain a speed of 15 m.p.h.
A pendulum consists of a thin straight uniform rod of mass \(M\) and length \(2l\) swinging about a certain point \(P\) of itself. It is found that when a particle of mass \(m\) is attached to the lowest point of the rod the period \(T\) of the pendulum does not alter, whatever the value of \(m\). Find the position of \(P\). The rod with the particle attached is now allowed to swing about a point of itself a small distance \(\xi\) above \(P\). Show that the period \(T'\) will change by the amount \(\dfrac{3\xi}{4l}\dfrac{2m-M}{4m+M}T\) approximately.
A particle can move on a smooth plane inclined at an angle \(\alpha\) to the horizontal and is attached to a point of the plane by a light inextensible string of length \(l\). The particle is at rest in equilibrium when it is given velocity \(V\) in the horizontal direction in the plane. Find the limits within which \(V\) must lie if the string is to become slack during the subsequent motion.
Three light rods \(BC, CA, AB\) each of length \(a\) are jointed together to form an equilateral triangle, which is suspended from a point \(O\) by three equal strings \(OA, OB, OC\) each of length \(b\). A weight \(W\) is suspended below the framework by three strings attached to \(A, B\) and \(C\) each of length \(c\). Find the resulting thrust created in each rod.
Explain what is meant by the angle of friction between two bodies in contact. The faces of a double inclined plane both make the same angle with the horizontal, and a rough heavy uniform chain lies across the ridge and stretches down lines of greatest slope on the two sides of the ridge. Find, in terms of the angle of friction, the inclination to the horizontal of the line joining the two ends of the chain if the system is in limiting equilibrium.
Show that referred to suitable axes the equation of the form in which a uniform heavy chain hangs under gravity is \(y=c\cosh\dfrac{x}{c}\). Obtain also the relation between \(x\) and the arc length \(s\). One end \(A\) of a uniform heavy chain \(AB\) of length \(2l\) is fixed while the other end \(B\) is moved very slowly along a horizontal rail through \(A\). Show that the locus described by the lowest point of the catenary in which the chain hangs, referred to horizontal and vertical (downwards) axes at \(A\), has equation \[ 2xy = (l^2-y^2)\{\log(l+y)-\log(l-y)\}. \]
A uniform perfectly rough plank of thickness \(2b\) rests across a fixed cylinder of radius \(a\) whose axis is horizontal. The plank is in equilibrium with its length inclined at an angle \(\alpha\) to the horizontal, and is rolled on the cylinder so that this angle increases to \(\theta\). Calculate the increase in potential energy of the system and hence show that the original position is one of stable equilibrium if \(a\cos^2\alpha>b\).
A particle of unit mass is allowed to fall from rest under gravity in a medium that produces on it a retardation equal to \(k\) times its velocity, and at the instant of release an equal particle is projected vertically downwards from the same place with initial speed \(v\). Show that their vertical distance apart tends ultimately to the value \(v/k\).