Find the locus of points \(P\) in the plane of a triangle \(ABC\) such that three forces through \(P\) whose lines of action pass through \(A, B, C\) and whose magnitudes are proportional to \(BC, CA\) and \(AB\) respectively can be in equilibrium.
A uniform circular hoop hangs in contact with a smooth vertical wall over a thin nail, which is perpendicular to the wall. A horizontal force \(P\) parallel to the wall acts on the hoop at the other end of the diameter through the nail. Show that equilibrium is possible for all values of \(P\) provided that the coefficient of friction between the nail and the hoop is not less than \[ \frac{1}{2\sqrt{2}}. \]
A uniform rod \(AB\) of length \(2b\) rests on the rim and inner surface of a smooth hollow hemispherical bowl of internal radius \(a\), which is fixed with the plane of its bounding circle horizontal. The end \(A\) of the rod is inside the bowl and a point \(C\) of the rod is in contact with the rim of the bowl. Find the reactions at \(A\) and \(C\) and show that \(b\) must lie between \(a\sqrt{2}\) and \(2a\).
A uniform rod \(AB\) of weight \(w\) and length \(a\) is smoothly hinged at \(A\) and is free to move in a vertical plane with the end \(B\) connected by a light elastic string of natural length \(a\) and modulus \(\lambda w\) to a fixed point, which is at a distance \(2a\) vertically above \(A\). Find the potential energy of the system when the extension of the string is \(x\). Show that if \(\lambda > \frac{1}{2}\) there are positions of equilibrium in which the rod is not vertical, and discuss the stability of all the positions of equilibrium for all values of the modulus.
The motion of a particle in a straight line is represented by a graph in which the velocity \(v\) is plotted against the displacement \(x\). If \(P\) is a point on the graph, \(M\) is the foot of the perpendicular from \(P\) on to the \(x\)-axis, and \(N\) is the intersection of the normal at \(P\) with the \(x\)-axis, show that the length \(MN\) represents the acceleration.
A particle moves along a straight line \(Ox\) so that when it is at a distance \(x\) from \(O\) its acceleration is \(n^2x\) directed away from \(O\). The particle is projected towards \(O\) with velocity \(nb\) from a point at a distance \(a\) from \(O\). Discuss the motion and sketch the \((v,x)\) graphs for the cases \(b>a, b=a, b
A train of total mass \(M\) pounds runs on a horizontal track, the frictional resistance being negligible. The engine can exert a maximum tractive force of \(P\) poundals, but cannot work at a rate exceeding \(Pu\) foot-poundals per second. Show that, if the train starts from rest, it cannot attain a speed of \(2u\) feet per second in less than \(\frac{5Mu}{2P}\) seconds, and that it cannot attain this speed in a run of less than \(\frac{17Mu^2}{6P}\) feet.
A particle is attached by a light elastic string of natural length \(a\) to a fixed point \(O\) from which the particle is allowed to fall freely. When the particle reaches its lowest position \(A\) the length of the string is \(3a\). Show that the time of falling from \(O\) to \(A\) is \[ \left(1+\frac{\pi}{3\sqrt{3}}\right)\sqrt{\frac{2a}{g}}. \]
A shell explodes on the surface of horizontal ground. Earth is scattered in all directions with varying velocities up to a certain maximum and falls over a circular area of radius \(2a\). Prove that the interval during which earth falls upon a spot on the ground at a distance \(a\) from the place of explosion may be as long as \(2\sqrt{\frac{a}{g}}\).
Find the acceleration of a particle which moves on a fixed circle of radius \(a\) with varying speed \(v\). \(OA\) and \(AB\) are two inextensible strings each of length \(a\). \(O\) is attached to a fixed point and masses \(9m\) and \(m\) are attached at \(A\) and \(B\). The system swings round so that both strings lie in a vertical plane which rotates about the vertical through \(O\) with constant angular velocity \(\omega\), the points \(A\) and \(B\) describing circles of radii \(x\) and \(3x\) respectively. Find the possible values of \(x\) and show that \(3aw^2 = g\sqrt{7}\) or \(15aw^2=8g\sqrt{10}\) according as the particles are on the same side or on opposite sides of the vertical through \(O\).
A uniform rod \(AB\) of length \(2a\) and mass \(m\) is balanced vertically on a smooth horizontal table, with the end \(A\) in contact with the table. A horizontal impulse \(I\) which is greater than \(\frac{2m}{3}\sqrt{(ag)}\) is applied at \(A\). Show that the rod will leave the table, and find the value of \(I\) for which the rod will land flat on the table. Does the rod leave the table if \(3I=m\sqrt{(ag)}\)?