It is required to bring to rest a weight \(W\) which has fallen freely from a height \(h\) by means of the direct pull of a rope of modulus \(\lambda\) one end of which is attached to it and the other to a point at a variable height vertically above. Find the minimum length of rope, if the tension is not to exceed a given value \(T\). Shew that with this length of rope the distance in which the weight is stopped is \[ \frac{2Wh}{T-2W}. \]
Establish the existence of the instantaneous centre of rotation (i.e. the point of no velocity) and the point of no acceleration for a rigid lamina moving in its own plane. Given these points and the angular velocity and acceleration of the lamina, determine the velocity and acceleration of any point. Shew that the locus of points of the lamina which are passing inflexions of their paths is a circle.
Draw the graph of \[ y = \frac{(x-a)(x-4a)}{x-5a}. \] Find the maximum and minimum values of \(y\).
By taking logarithms, or otherwise, find the limits of the positive value of \(\left(1+\frac{1}{x}\right)^x\) as \(x\) tends (a) to zero, (b) to infinity, positive values only of \(x\) being considered throughout. Draw roughly the graph of the function.
Two smooth elastic spheres (coefficient of restitution \(e\)) impinge obliquely in any manner; one of them being initially at rest, it is found that the angle between their subsequent directions of motion is constant. Find the ratio of the masses of the spheres, and the angle.
Discuss the motion of a particle in a uniform field of acceleration, and in particular the possibility of one or more paths through two given points. A bomb is dropped from an aeroplane at the correct moment to hit a certain object on the ground, on the assumptions that the aeroplane is at height \(h\), is moving in a vertical plane through the object, and is moving horizontally, and that its speed is \(v\). Neglecting the resistance of the air, determine the effect on the position of the point on the ground reached by the bomb, and on the time of flight, of small errors in each of these four assumptions.
The area of a triangle \(ABC\) is calculated from the measured values \(a, b\) of the sides \(BC, CA\) and the measured value \(90^\circ\) of the angle \(C\). It is found that the calculated area is too large by a small error \(z\), and that the true lengths of the sides are \(a-x, b-y\), where \(x\) and \(y\) are small. Shew that the error in the angle \(C\) is approximately \(\frac{180}{\pi} \left( \frac{2z-ay-bx}{\frac{1}{2}ab} \right)\) degrees.
Sum the infinite series:
The resistance to an airship is proportional to the square of the speed. It is required to cover a fixed distance in a fixed time. Shew that the work done is a minimum when the speed is constant.
Define Simple Harmonic Motion, and establish its chief properties. Discuss the result of compounding simple harmonic motions (1) in the same straight line, (2) in perpendicular straight lines, the periods being equal, (3) in perpendicular straight lines, one period being twice the other.