A plane uniform lamina is bounded by a semicircle of radius \(a\). Find its centre of gravity. A second plane uniform lamina, of different surface density, is bounded by a square of side \(2a\). A composite plane lamina is formed by joining the base of the semicircular lamina to a side of the square one. What is the ratio of the surface densities if the centre of gravity of the composite lamina is at the mid-point of the common edge?
One end of a uniform rod of weight \(w\) and length \(5l\) is freely hinged, while the other is attached by a light elastic string of unstretched length \(2l\sqrt{2}\) to a point at the same level as the hinge and distant \(7l\) from it. In equilibrium the length of the string is \(3l\sqrt{2}\). A weight \(W\) is now attached to the mid-point of the rod, and the length of the string in the new equilibrium position is \(4l\sqrt{2}\). Show that \(W=5w/3\).
A particle is projected under gravity with initial velocity \(v\) from a point \(O\) at a height \(h\) above a horizontal plane and strikes the plane at a horizontal distance \(d\) from \(O\). Find \(d_1\), the maximum value of \(d\), and, if \(d_2\) is the corresponding maximum distance when the point of projection is at a depth \(h\) below the plane, prove that \[ \frac{d_1^2}{d_2^2} = \frac{v^2+2gh}{v^2-2gh}. \]
A smooth sphere collides with a second smooth sphere with the same mass which is at rest; the coefficient of restitution is \(e\), and \(\theta\) is the angle just before impact between the direction of motion of the first sphere and the line of centres. Show that the change in the direction of motion of the first sphere is a maximum if \(2\tan^2\theta=1-e\), and explain this result if \(e=1\).
A bead of mass \(m\) moves under gravity on a smooth wire in the form of a parabola with its axis vertical and vertex uppermost. Prove that the pressure on the wire is \(mkg/ \rho\) when the bead is at a point where the radius of curvature of the wire is \(\rho\), and determine the constant \(k\) if \(4a\) is the latus rectum of the parabola and \(v\) is the velocity of the bead when it is at the highest point of the wire.
Two gravitating particles, of masses \(m_1, m_2\), are moving freely in a plane under their gravitational attraction which is of magnitude \(km_1m_2/r^2\) when the particles are at a distance \(r\) apart. If throughout the motion the particles are at a constant distance \(d\) apart, find the angular velocity of the straight line joining them.
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 \(x=a(\theta+\sin\theta), y=a(1-\cos\theta)\)].