Find the radial and transverse components of the acceleration of a point moving in a plane and whose position at any time is described by polar coordinates \(r, \theta\). A fine string is being unwound from a flat circular reel and the free portion \(PT\) is kept taut and in the plane of the reel. Show that if the point of contact \(T\) moves with constant angular velocity round the reel, the acceleration of the end \(P\) is always towards a certain point on the (moving) radius through \(T\), and determine its magnitude.
A heavy tube \(ABC\) is bent at right angles at \(B\) and the part \(AB\) is horizontal and slides freely through two fixed rings while the part \(BC\) is vertical. Two particles \(P\) and \(Q\) each of mass equal to that of the tube move in \(AB\) and \(BC\) respectively and are connected by a light inextensible string that can slide freely on the inside of the tube. If the system is released from rest, find the velocity of \(Q\) when it has descended a distance \(y\). Show also that the horizontal and vertical components of acceleration of \(Q\) are in the ratio 1:3.
A heavy flywheel which is known to be rotating with average angular velocity \(p\) is being driven by a variable driving couple \(G\sin^2 pt\) and retarded by a constant torque \(\frac{1}{2}G\) opposing the motion. Find the least moment of inertia that the flywheel must possess in order that the difference between its greatest and least angular velocities shall be less than \(p/N\), where \(N\) is a given large number.
A particle moves under an attraction varying inversely as the square of the distance from a fixed centre, and is describing a circle with period \(T\). Show that, if it is suddenly stopped and then allowed to fall freely, it will reach the centre of force after a time \(T/4\sqrt{2}\).
A uniform straight tube of mass \(M\) rests freely on a smooth horizontal table and contains a particle of mass \(m\) at rest at its mid-point. The system is set rotating about a vertical axis through its centre with angular velocity \(\omega\). Show that if the particle is slightly disturbed it will eventually leave the tube, and that at the instant it does so the angular velocity of the system will be \[ \frac{M+m}{M+4m}\omega. \]