A particle moves in a straight line under the action of a force towards a fixed point in the line and varying as the distance from the point. Find the position of the particle at any instant, having given its initial position and velocity. A heavy particle of mass \(m\) is attached to the end of an elastic string of natural length \(a\) and modulus \(\lambda\), the other end of the string being fixed to a point \(A\). The particle is released from rest at \(A\) and falls under gravity: prove that the string will be extended during the interval of time \[ 2\{\pi - \tan^{-1}(2\lambda/mg)^{1/2}\}/(\lambda/ma)^{1/2}. \]
\(O\) is the circumcentre and \(P\) is the orthocentre of a triangle \(ABC\). Prove that the resultant of forces completely represented by \(AP, BP, CP\) is completely represented by \(2OP\).
A uniform isosceles triangle \(ABC\) rests with its plane vertical and its two equal sides \(AB, AC\) in contact with two smooth fixed pegs \(P\) and \(Q\). \(PQ\) is horizontal. Prove that the angle between \(BC\) and \(PQ\) is either zero or \[ \sin^{-1}\left[\frac{BC}{6PQ}(1+\cos A)\right]. \]
Two uniform rods \(AB, BC\) of equal weight but different lengths, are freely jointed together at \(B\) and placed in a vertical plane over two equally rough fixed pegs in the same horizontal line. The inclinations of the rods to the horizontal are \(\alpha, \beta\), and they are both on the point of slipping. Prove that the inclination \(\theta\) to the horizontal of the reaction at the hinge is given by \[ 2\tan\theta = \cot(\beta+\lambda) - \cot(\alpha-\lambda), \] where \(\lambda\) is the angle of friction at the pegs.
A mass of 160 lb. is attached to one end of a light rope, the other end of which is made fast at a point \(A\). The rope is elastic, obeying Hooke's law, and its breaking tension is 2000 lb. wt. If the rope does not break when the mass is dropped freely from \(A\), prove that the elongation of the rope under its breaking tension must exceed 19 per cent.
A tug leaves a port to intercept a liner, which is proceeding with uniform velocity \(u\) miles per hour on a straight course which, at the nearest point, is \(a\) miles from the port. The tug starts when the liner is \(b\) miles from the port and has not yet reached the nearest point. Prove that the least uniform speed the tug must have in order to reach the liner is \(\frac{au}{b}\). Prove also that if the tug can go \(v\) miles per hour \((u > v > \frac{au}{b})\), the liner is on a part of her course in which the tug can intercept her for \(\frac{2\sqrt{b^2v^2-a^2u^2}}{u^2-v^2}\) hours.
A rocket is fired vertically from the surface of the earth, and it may be assumed that when it has risen to a height of 50 miles the charge is all expended and air resistance has become negligible. Prove that at this altitude the rocket must have a velocity of nearly 7 miles per second if it is to escape from the attraction of the earth. Assume that the radius of the earth is 4000 miles, and that the attraction of the earth varies inversely as the square of the distance from its centre. Neglect the motion of the earth.
Distinguish between the time-average and the space-average of a varying force acting on a moving body. The force acting on a body of mass 1 lb., which is initially at rest, varies as the square of the time, and is 10 lb. weight at the end of 10 seconds. Neglect resistances and gravity. Prove that the time-average and the space-average of the force during the first 10 seconds are \(3\frac{1}{3}\) lb. wt. and \(6\frac{2}{3}\) lb. wt. respectively.
Find \(\frac{dy}{dx}\) in the following cases:
The volume of water flowing uniformly per second down a given pipe (not quite full) of uniform circular section and uniform slope is proportional to \(\frac{A^{3/2}}{P^{1/2}}\), where \(A\) is the area and \(P\) is the curved portion of the perimeter of the cross-section of the stream. \(P\) subtends a (re-entrant) angle \(\theta\) at the centre of the section of the pipe. Prove that the volume flowing per second is a maximum when \[ 2\theta + \sin\theta = 3\theta\cos\theta. \]