A machine gun of mass \(M\) stands on a horizontal plane and contains a shot of mass \(M'\). The shot is fired horizontally at the rate of mass \(m\) per unit time with velocity \(v\) relative to the gun. If the coefficient of friction between the gun and the plane is \(\mu\), and sliding begins at once, show that the velocity of the gun after all the shot is fired is \begin{align} u\log\left(1 + \frac{M'}{M}\right) - \frac{\mu gM'}{m}. \end{align}
A flexible chain is in the form of a plane curve, and on each element \((s, s + ds)\) the distance measured along the curve from a fixed point, there are acting a tangential force \(R(s)ds\). If \(T(s)\) is the tension at \(s\) and \(\psi(s)\) is the inclination of the tangent at \(s\) to a fixed line, show that in equilibrium: \(F = dT/ds\), \(R = T(d\psi/ds)\), or otherwise, explain in the case of a uniform heavy chain hanging under gravity. In a vertical plane over a smooth cylinder of any convex section with axis perpendicular to the plane of the chain why the two free ends must be at the same horizontal level. Explain also why in the case of a light string hanging in a similar way over a similar but rough cylinder, and supporting weights on either side, equilibrium is possible only if the ratio of the weights on either side is less than a certain value, and find this limiting value in terms of the coefficient of friction between the string and the cylinder.
A light uniform (slightly flexible) beam of length \(l\) rests with its ends on two supports at the same horizontal level and at a distance \(l\) apart. A weight \(W\) is suspended from a point at a distance \(a\) from one end. With the usual assumption that the bending moment at any point of the beam is given by \(EIK\) where \(K\) is the curvature, show that the downward deflection of the beam at the point from which the weight is suspended is given by $$\frac{1}{6}(W/EI)a^2(l-a)^2$$ approximately. Find the point of the beam where the downward deflection is greatest, and verify that it is in the larger segment of the beam.
A rigid body consists of a thin heavy rigid wire in the shape of a circle of radius \(a\) and centre \(O\), and a heavy particle of negligible dimensions attached to the wire at a point \(P\) of its circumference. The body is suspended freely from a point \(Q\) of the circumference which can be varied at will. The body executes small oscillations in plane of the circle about the position of stable equilibrium. If \(l\) is the length of the equivalent simple pendulum, show that the least value of \(l\) is given by \(2a(1+2\lambda)\frac{1}{2}(1+\lambda)^{-1}\), where \(\lambda\) is the fixed ratio of the mass of the particle to that of the wire, and find the value of the angle \(POQ\) to give this value. Find also the greatest value of \(l\) and the corresponding values of the angle.
A bead of mass \(m\), which is free to move on a smooth wire in the form of an ellipse held fixed in a vertical plane with its major axis vertical, is attached to one end of a light elastic string of modulus \(\lambda\) whose other end is attached to the uppermost focus of the ellipse, so that when the particle is at the top end of the major axis of the ellipse the string is just taut. Find the possible positions of equilibrium, and show that there is an oblique position if $$\frac{\lambda}{mg} > \frac{1-e}{2e^2},$$ where \(e\) is the eccentricity of the ellipse.
A bead of unit mass is projected with horizontal velocity \(u\) at the vertex of a smooth rigid parabolic wire held fixed in a vertical plane with its axis vertical and its vertex uppermost and moves under gravity on the wire. Prove that when the bead is at depth \(y\) below the vertex, the pressure on the wire is given by $$\left(\frac{a}{a+y}\right)^{\frac{1}{2}}\left(g-\frac{u^2}{2a}\right),$$ where \(2a\) is the length of the semi-latus rectum of the parabola. Explain what happens when \(u^2 = 2ga\). Show also that if the wire is made to terminate at any point and the bead allowed to fly off at a tangent, the resulting path is a parabola with the same directrix whatever the point at which the bead leaves the wire. Find the position of this directrix.
A particle is projected from a point \(O\) with velocity having components \(u\) and \(v\) horizontally and vertically upwards respectively, and moves under gravity and a resisting force per unit mass of \(k\) times the square of the resultant velocity in the reverse direction of this velocity. Show that at the point on the path at a distance along the path of \(s\) from \(O\), the horizontal component of the velocity is \(ue^{-ks}\). By considering the special case when \(u\) is zero, show that in general the vertical height attained above \(O\) cannot exceed $$\frac{1}{2k}\log_e\left(1+\frac{kv^2}{g}\right).$$
A particle of unit mass moves under an attractive force \(f(r)\) directed towards a fixed point \(O\). If \(v\) is the velocity of the particle and \(p\) the perpendicular from \(O\) on to the tangent to the path at any point \(P\), show that \(vp = h\), a constant: and that $$\frac{h^2 dp}{p^3 dr} = f(r),$$ where \(OP = r\). Hence, or otherwise, show that if in the special case when \(f(r) = \mu r^2\), the particle is projected when at a great distance from \(O\) with a velocity \(v_0\) along a straight line at a small perpendicular distance \(c\) from \(O\), in the subsequent motion the distance of closest approach of the particle to \(O\) is given approximately by \(v_0^2c^2/2\mu\).
A rigid smooth straight thin tube is made to rotate in a vertical plane with angular velocity \(\omega\) about a fixed point \(O\) of itself. A particle can move freely inside the tube is released when at relative rest at a distance \(a\) from \(O\) when the tube is tangentially horizontal. Obtain an expression for the distance \(r\) of the particle from \(O\) at subsequent time \(t\) and verify that in the special case \(2\omega^2 = g\), the motion from \(O\) is with indefinite increase of time to a simple harmonic motion of period \(2\pi \omega^{-1}\) about \(O\), and that the tube may be assumed to be sufficiently long to contain the particle during its motion.
A smooth uniform stationary sphere of mass \(m\) is hit obliquely by a similar sphere, mass \(m_1\) whose velocity is in a direction inclined at an angle \(\alpha\) to the line of centres of the spheres at impact. Prove that after impact the directions of the velocities of the spheres are inclined at an angle \(\beta\) where \((m_1 + m_2)\tan\alpha = (m_1 - cm_2)\tan\beta\), \(c\) being the coefficient of restitution. Explain what happens when \(m_1 = cm_2\). In the case of a perfectly elastic collision (\(c = 1\)) show that when \(m_1 > m_2\), the greatest deviation which can be produced by the collision in the direction of the velocity of \(m_1\) is where \((m_1^2 - m_2^2)^{\frac{1}{2}}\tan\delta = m_2\).