10273 problems found
A particle is projected with velocity \(V\) from a point on an inclined plane in such a way that when it reaches the plane again it strikes it at right angles. Show that the range on the plane is \[ \frac{V^2}{g} \frac{2\sin\alpha}{1+3\sin^2\alpha}, \] where \(\alpha\) is the angle which the plane makes with the horizontal.
A train can be accelerated by a force of 55 lb. weight per ton, and, when steam is shut off, can be braked by a force of 440 lb. weight per ton. Find the least time between stopping points 3850 ft. apart; find the greatest velocity attained and the horse-power per ton weight required at this greatest velocity.
A light inelastic string passes over two light pulleys which lie in the same vertical plane. Between the pulleys, the string hangs vertically and passes under a third light pulley carrying a load of mass \(m_3\). The free ends of the string hang vertically and carry masses \(m_1\) and \(m_2\). If the system is released from rest, show that the downward acceleration of the centre of gravity of the three masses is \[ g \left[ 1 - \frac{16m_1 m_2 m_3}{(m_1+m_2+m_3)(4m_1 m_2 + m_2 m_3 + m_3 m_1)} \right]. \]
A uniform smooth spherical ball of mass \(m\) suspended by a light inextensible string from a fixed point hangs at rest under gravity, and a similar ball falls vertically and strikes the first when travelling with velocity \(v\). At the instant of impact the line of centres of the spheres makes an angle \(\alpha\) with the vertical. Prove that the loss of energy at impact is \[ \frac{1}{2}mv^2 \frac{1-e^2}{1+2\tan^2\alpha}, \] where \(e\) is the coefficient of restitution.
Two particles of masses \(2m\) and \(m\) are attached to the ends of a light elastic flexible string of natural length \(a\) and modulus \(mg\), which is free to slip over a smooth peg. Initially the mid-point of the (unstretched) string is held in contact with the peg, and the particles hang in equilibrium on either side of the peg. The mid-point is now set free. If \(x\) and \(y\) denote the depths of the particles below the peg at time \(t\) subsequent to the release, show that \[ 2\ddot{x}-\ddot{y} = g, \quad \ddot{x}+\ddot{y} = \frac{7}{2}g - \frac{3g}{2a}(x+y). \] Verify that \begin{align*} x &= \frac{13a}{9} - \frac{1}{6}gt^2 + \frac{a}{18}\cos nt, \\ y &= \frac{8a}{9} - \frac{1}{6}gt^2 + \frac{a}{9}\cos nt, \end{align*} where \(n^2=3g/(2a)\), these expressions being valid until one particle reaches the peg.
Show that when a particle describes a curve its acceleration components along and perpendicular to the curve are \(\frac{d v}{d t}\) and \(\frac{v^2}{\rho}\), where \(v\) is the velocity of the particle, and \(\rho\) is the radius of curvature at the instantaneous position of the particle. \par Two equal particles are connected by a light inelastic string of length \(\pi a\), and are placed on a smooth circular cylinder of radius \(2a\) which has its axis horizontal, so that they rest in unstable equilibrium with the string passing over the top of the cylinder. If the equilibrium is slightly disturbed, find the tension in the string and the reactions of the particles on the cylinder in terms of the angular displacement, and show that the lower particle leaves the cylinder when at an angular distance of approximately 77\(^\circ\) 20' from the top.
Two conics \(S\) and \(S'\) meet in the four points \(A, B, C, D\). Through \(A\) a variable line \(l\) is drawn meeting \(S\) in \(P\) and \(S'\) in \(X\). \(BP\) meets \(S'\) again in \(Y\), and \(BX\) meets \(S\) again in \(Q\). Show that if \(A, Y, Q\) are collinear for one position of \(l\), then this will also be the case for every position of \(l\), and as \(l\) varies, the lines \(PQ\) all pass through a fixed point \(U\), and the lines \(XY\) all pass through a fixed point \(V\). Show further that in this case the pole of \(AB\) with respect to \(S\) is the same as the pole of \(CD\) with respect to \(S'\), and the pole of \(CD\) with respect to \(S\) is the same as the pole of \(AB\) with respect to \(S'\).
Deduce from the triangle (or parallelogram) of forces (i) that a system of forces in a plane can be reduced either to a single force or to a couple, (ii) that the sum of the moments of the forces about any point in the plane is equal to the moment of this resultant force about the point or, if the system reduces to a couple, to the moment of this couple.
Prove that the inverse of a circle is either a circle or a straight line. Two fixed circles \(C\) and \(C'\), of radii \(r\) and \(r'\), touch at the point \(O\). Two variable circles \(\Gamma\) and \(\Gamma'\) each touch both \(C\) and \(C'\), and also touch each other at the point \(P\). Show that as \(\Gamma\) and \(\Gamma'\) vary, \(P\) describes a circle touching \(C\) and \(C'\) at \(O\). Find the radius of this circle in terms of \(r\) and \(r'\), distinguishing between the cases in which \(C\) and \(C'\) are on the same or on opposite sides of their common tangent.