An arc of a circle formed of thin uniform wire hangs at rest under gravity from a point \(P\) of the arc; \(Q\) is the point of the circle (not necessarily on the arc) vertically below \(P\). If the wire oscillates freely about \(P\) in a vertical plane through \(P\), prove that the equivalent simple pendulum is of length \(PQ\).
Describe briefly the geometrical process by which the resultant of two forces at a point can be found, and show from the process how three non-coplanar forces at a point can similarly be combined. A uniform triangular plate is suspended from a fixed point by means of three strings attached to the vertices. Prove that the tensions in the strings are proportional to their lengths.
A heavy uniform solid hemisphere rests in equilibrium with its curved surface in contact with a horizontal plane and a vertical wall, and is symmetrically situated so that the plane face is parallel to the line in which the wall and plane meet. Show that, if \(\mu\) is the coefficient of friction at the ground and \(\mu'\) that at the wall, the greatest inclination \(\theta\) of the plane face to the horizontal is given by \[ \sin\theta = 8\mu(1+\mu')/3(1+\mu\mu'), \] provided the value of this expression does not exceed unity. Discuss briefly the case when it does exceed unity.
Show that for the form of any chain of continuous line density hanging under gravity between two fixed points, the intrinsic equation must be the integral of the equation \[ w\rho\cos^2\psi = \text{constant}, \] where \(w\) is the weight per unit length at a general point at which \(\rho\) is the radius of curvature and \(\psi\) the inclination of the tangent to the horizontal. Find the law of density (\(w\) as a function of \(\psi\)) in order that a chain may hang (i) as a parabola with axis vertical, (ii) in the form of a cycloid [that is, the curve whose parametric Cartesian equation is \(x=a(\theta+\sin\theta), y=a(1-\cos\theta)\)].
A uniform perfectly rough heavy plank of thickness \(2b\) rests symmetrically across the top of a fixed horizontal circular cylinder of radius \(a\), the length of the plank being perpendicular to the axis of the cylinder. Show that if the plank is turned without slipping through an angle \(\theta\) round the direction of the axis of the cylinder, the gain of potential energy is proportional to \[ a\theta\sin\theta - (a+b)(1-\cos\theta), \] and find the condition that the horizontal position shall be stable. Show further that if \(a>b\) there always exists a second position of equilibrium into which the plank can be turned.
A gun of mass \(M\) is free to recoil on a horizontal plane, and a shell of mass \(m\) is fired from it with the barrel elevated at an angle \(\alpha\). Show that if the muzzle velocity of the shell in space is \(v\), the horizontal range will be \[ 2v^2\beta/g(1+\beta^2), \] where \(\beta=(1+m/M)\tan\alpha\) and \(g\) is the acceleration of gravity.
A smooth sphere rests on a horizontal plane and is in contact with an inelastic vertical plane. An equal sphere moving on the horizontal plane with velocity \(v\) in a direction perpendicular to the wall strikes the first sphere and at impact the line of centres makes an angle \(\theta\) with the direction of \(v\). Find the speed communicated to the first sphere in terms of \(\theta\), and show that its greatest possible value is \(v(1+e)/2\sqrt{2}\), where \(e\) is the coefficient of restitution between the spheres.
A heavy particle of mass \(M\) rests on a smooth horizontal table at the centre of an equilateral triangle of side \(2a\), and three other particles each of mass \(m\) are attached to it by three strings that pass through holes in the table at the vertices of the triangle. The three particles hang vertically and initially the system rests in equilibrium. If one of the strings is suddenly cut, find the instantaneous change of tension in the other two and show that \(M\) begins to move with acceleration \(2mg/(2M+m)\). Find also the velocity of \(M\) when it crosses the side of the triangle.
A simple pendulum is making complete revolutions in a vertical plane in such a way that its greatest and least angular velocities are \(\omega_1\) and \(\omega_2\). Show that when the inclination of the pendulum to the downward vertical is \(\theta\) the angular velocity is \[ (\omega_1^2 \cos^2 \theta/2 + \omega_2^2 \sin^2 \theta/2)^\frac{1}{2}. \] Find the corresponding formula for the tension in terms of its greatest and least values, and hence show that critical values of the tension can never occur except at the highest and lowest points.
A straight rod \(OQ\) of length \(a\) rotates round \(O\) with constant angular velocity \(\omega\) so that \(Q\) describes a circle, and a rod \(QP\) of length \(b\) is freely jointed at \(Q\), while \(P\) is constrained to move in a straight line through \(O\). Show that when \(OQP\) are collinear, with \(Q\) between \(O\) and \(P\), the acceleration of \(P\) is of magnitude \(a(a+b)\omega^2/b\). Find also the acceleration of \(P\) when \(OQ\) is at right angles to \(OP\).