Problems

It is desired that the performance of a model of a machine should correspond with that of the machine itself. Explain how its masses, linear dimensions, and speed, and the forces applied to it, must be adjusted in comparison with those of the machine.

Solve the equations

1. [(i)] \(4 \cos\theta \cos 2\theta \cos 3\theta - 1 = 0\),
2. [(ii)] \(4 \cos\theta \cos 2\theta \cos 5\theta + 1 = 0\).

Two particles of mass \(M\) and \(m\) (\(M>m\)) are placed on the two smooth faces of a light wedge which rests on a smooth horizontal plane. The faces of the wedge are inclined to the horizontal at angles \(\alpha\) and \(\beta\), respectively. If the system starts from rest, shew that the smaller particle will move up the face on which it is placed if \[ \tan \beta < \frac{M \sin\alpha \cos\alpha}{M \sin^2\alpha+m}. \]

Solve the equations:

1. [(i)] \(\cos x + \sin x = 1\),
2. [(ii)] \(\cos x + \cos 3x + \cos 5x = 0\).

State the principle of virtual work, and give a proof of it for the case of a single rigid body. What types of force do not appear in the equation of virtual work? Explain how the work function may be used to determine the stability of a system. \par Two equal particles repel each other according to the fifth power of the distance, and are connected by an elastic string. Find the position of equilibrium, and shew that it is stable if the extension of the string is less than one-quarter of its original length.

A frame of steel bars, in the form of a square and two diagonals, is suspended by one angle, a given weight being attached to the opposite angle. The sides of the square and the horizontal diagonal are of equal thickness, half that of the vertical diagonal. Find the tension of the vertical diagonal, neglecting the weight of the frame, and assuming that it has no stress when not loaded.

Prove that \[ \sum_{r=0}^{n-1} \frac{1}{1-\cos\left(\phi+\frac{2r\pi}{n}\right)} = \frac{n^2}{1-\cos n\phi}. \]

A body of mass one lb. is projected on a rough plane surface with a velocity of 10 feet per second, and its velocity after time \(t\) is given for various values of \(t\) by a smooth curve passing through the points defined by the following table:

Derive the curve connecting the retardation with the distance travelled, and by determining the area of this curve verify that the energy lost during the 30 seconds is 1.1 foot-lb. approximately.

Prove that the area bounded by the hyperbola \(xy=1\), the axis of \(x\), and the ordinates \(x=1\) and \(x=2\), is less than 1; and that similarly bounded by the ordinates \(x=1\) and \(x=3\) is greater than 1.

Give an account of the theory of the parabolic motion of a projectile under the influence of gravity only. In particular describe the use of graphical methods depending on the geometrical properties of the focus and directrix of the path. Prove that the paths of all particles projected with a given velocity from a given point, the direction being arbitrary, touch a surface formed by the rotation of a certain parabola about the vertical line through the point of projection.