Problems

A heavy particle is attached to the rim of a wheel of radius \(r\) which is made to rotate in a vertical plane with constant angular velocity \(\omega\) about its centre which is fixed. Shew that if the particle is set free at some point of its path the time \(t\) taken to reach the horizontal plane through the lowest point of the wheel is given by \(g^2 t^4 - 2 t^2 r (g+r\omega^2) + x^2 = 0\), where \(x\) is the horizontal distance traversed measured from the lowest point of the wheel. Deduce that the greatest value of \(x\) is \(r + \omega^2 r^2 / g\).

A particle moving in vacuo passes with a given velocity \(q\) through a fixed point \(O\). Shew that all possible paths are parabolas of which the directrices lie in a common horizontal plane; and use this result to determine, by means of a geometrical construction, the paths which contain a second specified point. If the path also passes through \(Q\), where \(OQ\) makes a given angle \(\alpha\) with the horizontal, determine the conditions under which the distance \(OQ\) will have its maximum value, and prove that, when these conditions are satisfied:

1. [(i)] the directions of motion at \(O\) and at \(Q\) are perpendicular to one another;
2. [(ii)] the time occupied in passing from \(O\) to \(Q\) is equal to the time which would be occupied in dropping from rest through a vertical distance equal to \(OQ\);
3. [(iii)] \(Q\) lies on a fixed paraboloid having its focus at \(O\).

Shew that the locus of the foot of the perpendicular from the centre of the ellipse \(x^2/a^2 + y^2/b^2 = 1\) on to a tangent is the curve \[ (x^2+y^2)^2 = a^2x^2+b^2y^2, \] and that the corresponding locus for the rectangular hyperbola \(xy = c^2\) is \[ (x^2+y^2)^2 = 4c^2xy. \]

Express \[ \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2 + \left(\frac{\partial u}{\partial z}\right)^2 \] in terms of \(r, \theta, \phi\), and the partial derivatives of \(u\) with respect to \(r, \theta, \phi\), where \[ x = r \sin\theta \cos\phi, \quad y = r \sin\theta \sin\phi, \quad z=r\cos\theta. \]

At speeds over 8 miles an hour, the total tractive force at the rims of the wheels of an 11 ton tramcar is given by the equation \(P(v-5) = 7000\), where \(P\) is the force in pounds weight and \(v\) is the velocity in miles an hour. Shew that the tramcar can accelerate from 8 to 12 miles an hour in about 16 yards.

Prove that in the motion of a system of particles in one plane:

1. [(i)] the increase of kinetic energy in any time is equal to the work done by the forces in that time;
2. [(ii)] the centre of mass moves as a particle of mass equal to that of the system under the action of all the forces moved parallel to themselves to act at the centre of mass;
3. [(iii)] the motion relative to the centre of mass is independent of the motion of that point.

A uniform rod of length \(l\) has one end fastened to a fixed point which is at a height \(l\) above the ground. The rod is given an initial angular velocity \(\omega\) in a vertical plane, starting from rest in its lowest position; and the fastening gives way when the rod first reaches a horizontal position. Shew that, if the rod is horizontal when it strikes the ground, then \[ l\omega^2 = 3g + gk^2\pi^2/(2+k\pi), \] where \(k\) is a positive integer.

If \[ y = \frac{x-1}{(x+1)^2}, \] shew that \(y\) can never be greater than \(\frac{1}{8}\). Sketch the graph; and find the point in which the line which joins the points in which the curve meets the axes meets the curve again.

\(A\) and \(B\) are two given points, and \(P\) a variable point on a given straight line parallel to \(AB\). Prove that (i) if the line cuts the circle on \(AB\) as diameter, \(AP \cdot BP\) is a maximum when \(P\) lies on the perpendicular bisector of \(AB\), and (ii) if the line does not meet the circle, \(AP \cdot BP\) is a minimum when \(P\) lies on this perpendicular bisector. Examine the case when the line touches the circle.

Two particles of masses \(m\) and \(m'\) are connected by a fine thread passing over a small smooth pulley at the top of a smooth fixed solid whose vertical section is a quadrant of a circle, as in the figure. If motion begins when the radius to the particle \(m\) is horizontal, shew that when the radius has turned through an angle \(\theta\) the pressure between the mass \(m\) and the surface is \[ mg\{(3m+m')\sin\theta - 2m'\theta\}/(m+m'), \] so long as this expression remains positive.

Find the \(n\)th differential coefficients of \[ \cos x, \quad \cos^2 x, \quad \log(1+x), \quad \frac{x}{1+3x+2x^2}. \]