Problems

A cycloid may be defined as the locus of a point on the rim of a wheel of radius \(a\), which rolls without slipping along a horizontal straight line, the plane of the wheel being vertical; from this definition prove that the coordinates of any point on the cycloid may be written in either of the forms:

1. [(i)] \(x = a(\theta - \sin\theta), \quad y = a(1-\cos\theta)\);
2. [(ii)] \(x = a(\phi + \sin\phi), \quad y = a(1-\cos\phi)\).

Explain and illustrate by a diagram the choice of coordinate axes in each case and state the relation between the parameters \(\theta, \phi\). The distance of a variable point P of a curve from a fixed point of the curve measured along the curve is \((A + B\cos\psi + C\sin\psi)\), where A, B, C are constants and \(\psi\) is the inclination of the tangent at P to a fixed line; prove that the curve is a cycloid.

[This question was too poorly scanned to be transcribed reliably.]

On a rough inclined plane are placed a uniform block in the shape of a rectangular parallelepiped, of weight \(W\), with an edge horizontal, and, above the block and resting against it with the point of contact at the centre of a face of the block, a uniform sphere of weight \(W\) and radius \(a\). The edges of the block parallel to a line of greatest slope are of length \(3a\); the other edges are of length \(2a\). The coefficient of friction at all pairs of surfaces in contact is \(\frac{1}{2}\). The inclination \(\alpha\) of the plane to the horizontal is gradually increased from a very small value. Shew that when equilibrium is first broken the sphere does not begin to slip down the plane with its point of contact with the block unchanged, but rolls down the plane and slips against the block; and that the block slides and does not tilt. Shew also that equilibrium is first broken when \(\tan\alpha = \frac{1}{4}\).

A smooth wire in the form of a circle is placed in a vertical plane, and a bead of weight \(W\) which slides on it is attached to the highest point of the circle by a weightless elastic string of modulus \(kW\), whose natural length is equal to the radius of the circle. Find (i) the condition that the lowest position of the bead should be one of stable equilibrium, (ii) the condition that other equilibrium positions should be possible, (iii) where the other equilibrium positions are and whether or not they are stable.

A particle is projected along the outside surface of a smooth sphere of radius \(a\) ft. from the highest point with velocity \(\frac{1}{2}\sqrt{(ga)}\). Prove that it strikes a horizontal plane through the centre of the sphere at a distance \[ \frac{9\sqrt{39} + 7\sqrt{7}}{64} a \text{ ft.} \] from the centre.

A particle of mass \(m\) moves in a straight line on a rough horizontal table, under the influence of the frictional force, which has the constant magnitude \(n^2ml\), and a force of magnitude \(n^2mx\) towards a point O in the line of motion, where \(x\) is the displacement from O. The particle is released from rest at a distance \(15.5l\) from O. Shew that it performs simple harmonic half-oscillations about each of two points in turn, and that it comes finally to rest after making an integral number of half-oscillations; and find how many half-oscillations it makes before it comes finally to rest.

Two equal masses are fixed to a light rod, one at the top point and one at the middle point, and the lower end of the rod rests on a rough horizontal table. The rod is released from rest at an angle \(\alpha\) to the vertical. Shew that the lower end will not slip initially, for any value of \(\alpha\), if the coefficient of friction exceeds \(9/(2\sqrt{10})\).

If \(x=\frac{2}{\sqrt{7}}\sin\theta\), express \(\frac{\sin 7\theta}{\sin\theta}\) as a polynomial in \(x\). Shew that the roots of the equation \[ x^3-x^2+\frac{1}{7}=0 \] are \[ \frac{2}{\sqrt{7}}\sin\frac{2\pi}{7}, \quad \frac{2}{\sqrt{7}}\sin\frac{4\pi}{7}, \quad \frac{2}{\sqrt{7}}\sin\frac{8\pi}{7}. \]

Prove that a system of coplanar forces is in general statically equivalent to two forces one of which is of given magnitude, direction and line of action. Forces of magnitudes 1, 2, 3, 4 act along the sides \(\vec{AB}\), \(\vec{BC}\), \(\vec{CD}\), \(\vec{DA}\) of a square \(ABCD\). The system is equivalent to two forces one of which is of magnitude 3 and acts along \(AC\); find the magnitude and direction of the second force, and find where its line of action cuts \(AB\), \(AD\).

Prove that, if the perpendiculars from \(A'\), \(B'\), \(C'\) to the sides \(BC\), \(CA\), \(AB\) of the triangle \(ABC\) are concurrent, then \[ A'C^2+B'A^2+C'B^2 = A'B^2+B'C^2+C'A^2. \] Prove further that the perpendiculars from \(A\), \(B\), \(C\) to \(B'C'\), \(C'A'\), \(A'B'\) are concurrent.