Problems

Two particles, \(A\), \(B\), of masses \(m_1\), \(m_2\) respectively, are connected by a light spiral spring, which obeys Hooke's law, and move on a smooth horizontal table along the line of the spring (supposed to remain straight). Write down the equations of motion of the masses in terms of their displacements \(x_1\), \(x_2\) and deduce that

1. the centre of gravity \(G\) of the system moves with constant velocity;
2. the distance \(x_2 - x_1\) between \(A\) and \(B\) varies harmonically;
3. the distance of either mass from \(G\) varies harmonically with the same period as \(x_2 - x_1\).

A rectangular window-sash of width \(a\) and height \(b\) slides vertically in equally rough grooves at its two vertical edges. The weight of the window is \(W\) and its centre of gravity is at its geometrical centre. The window is supported by two light vertical cords which are attached to its two upper corners and which, after passing over smooth light pulleys, each carry a weight \(\frac{1}{2}W\). The window has a little sideways play in its plane so that it can be moved up and down easily. While the window is open, one of the cords breaks but the window remains open. Prove that the coefficient of friction between the window and its grooves must be at least \(b/a\).

A tricycle has a light frame, two back wheels each of weight \(w\) and a front wheel of weight \(w'\). The tricycle stands on level ground with the back wheels touching the ground at points \(B\) and \(C\) distant \(2b\) apart. The front wheel touches the ground at a point \(A\), and the join of \(A\) to the mid point of \(BC\) is perpendicular to \(BC\) and of length \(a\). Prove that a man of weight \(W\) riding the tricycle with uniform velocity will not tip the tricycle over, if he keeps the vertical through his centre of gravity within a certain region of the ground of area \(ab (W + 2w + w')^2/W^2\).

Each side of a steep ramp is composed of eleven equal smoothly jointed light rods in a vertical plane as indicated in the diagram. The ramp is smoothly hinged at \(G\) and rests smoothly on a horizontal plane at \(A\); the inclination of the ramp to the horizontal is \(30^\circ\). A vehicle of weight \(W\) two-thirds of the way down the ramp produces a stress in the frame- work comprising either side of the ramp equal to that due to weights \(\frac{1}{6}W\) at \(A\), \(\frac{1}{3}W\) at \(C\) and \(\frac{1}{2}W\) at \(E\). Draw a stress diagram and find the stress in each member of the framework. % Diagram is omitted as per instruction limitations.

A string is in limiting equilibrium in contact with a normal section of a rough cylindrical surface (coefficient of friction \(\mu\)) and is under the action of no forces except the reaction of the surface and the tensions applied to its ends. Write down the equations of equilibrium of an element of the string, and find the way in which tension varies along the string. A torque amplifier consists of a circular drum which is made to revolve on its axis with uniform angular velocity \(\Omega\) and round which are wrapped \(n\) closely spaced turns of a thin light rough string as indicated in the diagram. The ends of the string are attached to the arms of a pair of cranks whose shafts are coaxal with the axis of the drum and on opposite sides of it. The arms of the cranks are each of length greater than the radius of the drum. The crank shafts are unconnected to each other or to the drum except via the string. The output shaft of the torque amplifier is used to drive, with angular velocity less than \(\Omega\), a mechanism which requires a torque \(G\) to operate it. Find the minimum torque that needs to be applied to the input shaft in terms of \(G, n\) and \(\mu\), the coefficient of friction between the string and the drum. All directions of rotation are left-handed about a line drawn from the input to the output shaft, but the string forms a right-handed spiral about this direction so that it remains taut. % Diagram is omitted as per instruction limitations.

A four-wheeled railway-truck has a horizontal floor and may be regarded as a rect- angular box of length \(2l\) and width \(2a\) with axles distant \(2d\) apart. The vertical planes through the geometrical centre of the box parallel and perpendicular to the axles are planes of symmetry for the whole structure, and the wheels together with their axles do not protrude outside the vertical planes containing the sides and ends of the box. The truck moves through a tunnel having vertical sides distant \(2b\) apart. The tunnel is curved and the vertical sides are portions of coaxal circular cylinders of radii \(R-b\) and \(R+b\). The track is laid in such a way that the mid-points of the axles trace out a horizontal circle of radius \(R\) with centre on the common axis of the cylindrical walls. Find the minimum value of \(b\) necessary to allow a clearance \(c\) between the truck and the tunnel. Show that, for tunnels of small curvature, the clearances on both sides of the tunnel are the same, if \(d^2 = \frac{1}{3}l^2\), and that in these circumstances the reduction in clearance due to the curvature of the tunnel is \(l^2/R\).

A uniform rectangular door of mass \(m\) and width \(a\) swings on a vertical axis at one edge and has a catch at the other. To make the door close and fasten automatically the hinges are so adjusted that, as the door is opened, it is also raised uniformly by an amount \(h\) per radian. To make the door fasten, an impulse \(P\) must be applied at the catch normal to the door. If the hinges are well oiled, so that friction may be neglected, shew that the door will close and fasten after being opened through an angle \(\alpha\) provided that \(\alpha > 3P^2/(2m^2gh)\). If, on the other hand, the door is stiff, so that a resistive couple equal to \(R\) times the angular velocity is brought into play, shew that the door will close and fasten provided that \(R

A tram is travelling with uniform velocity along a straight horizontal track, and the pivot of the trolley-arm is vertically under the conductor-wire, which may be taken as horizontal. The trolley-arm is a uniform straight rod of length \(l\) and weight \(W\), making an angle \(\alpha\) with the horizontal. The weight of the trolley-wheel is \(w\) and its bearings are smooth. A reaction of magnitude \(R\) is maintained between the trolley-wheel and the conductor-wire to ensure that the wheel remains on the wire. Find the distribution along the trolley-arm of thrust, shearing force and bending moment. State what couple must be maintained at the pivot.

The centre of gravity of a four-wheeled car is located between the axles at a height \(h\) above the road. The horizontal distances of the centre of gravity from the front and back axles are \(a\) and \(b\) respectively. The car is driven from the back wheels only and the bearings of the front wheels are smooth. Rotary inertia of the wheels may be neglected. The car is accelerated along a straight horizontal course, and the coefficient of friction between the wheels and the road is \(\mu\). Shew that the maximum possible acceleration without skidding is \[ \frac{\mu ag}{a+b-\mu h} \] provided that \(\mub/h\), the maximum safe acceleration is \(bg/h\).