A crane is built of light jointed bars as in the figure. Sketch the force diagram, showing which members are in tension or thrust, and find the reactions at the two points of support.
Explain the principle of virtual work. A tripod of three equal light rods of length \(l\), loosely jointed together at the top, rests on a smooth table, their lower ends being held together by three equal horizontal strings of length \(a\), which join them in pairs. A weight \(W\) is hung from the top. Find the tensions of the strings.
Explain the construction of the funicular polygon, showing in particular what it becomes when the system of forces (a) is in equilibrium, (b) reduces to a couple. Forces of magnitudes 1, 2, 3, 4, 5 act along the successive sides of a regular pentagon, taken round it in the same direction. Sketch figures to show the magnitude, direction and line of action of the resultant force.
A uniform ladder of weight \(W\) leans with one end against a wall and makes an angle \(\theta\) with the floor. The angles of friction for floor and wall are respectively \(\epsilon\) and \(\eta\). Explain why it is not in general possible to determine the reactions at the ends of the ladder. If a man of weight \(w\) slowly climbs the ladder, show that he can get to the top if \[ \frac{1}{2}\frac{W}{W+w} > \frac{\cos\eta \cos(\epsilon+\theta)}{\cos(\eta-\epsilon)\cos\theta}. \]
A particle is projected with velocity \(V\) at angle \(\alpha\) to the horizontal. Find the range and time of flight. A shell bursts on contact with the ground and pieces from it fly in all directions with all velocities up to 80 feet per second. Show that a man 100 feet away is in danger for \(1/\sqrt{2}\) seconds.
A string passes over a smooth fixed pulley and to one end there is attached a mass \(M_1\), and to the other a smooth light pulley over which passes another string with masses \(M_2\) and \(M_3\) at the ends. If the system is released from rest show that \(M_1\) will not move if \[ \frac{4}{M_1} = \frac{1}{M_2} + \frac{1}{M_3}. \] What is the pressure on the fixed pulley?
The velocity of a stream between parallel banks at distance \(2a\) apart is zero at the edges and increases uniformly to the middle where it is \(u\). A boat is rowed with constant velocity \(v\) (\(>u\)) relative to the water, and goes in a line straight across. How are the bows pointed at any point of the path and how long will it take to get across?
A flywheel of mass \(M\) is made of a solid circular disc of radius \(a\). Find its kinetic energy when it rotates \(n\) times a second. A ring of radius \(b\) is mounted on a shaft in line with the axis of the flywheel, and is driven by an engine at \(n'\) revolutions a second. It can be pressed against the flywheel so as to act as a clutch. If the pressure is \(P\) and the coefficient of friction \(\mu\), find how long it takes for the flywheel to get up full speed from rest, and find the rate at which the engine does work during the process.
Find the radial and transverse accelerations of a particle in polar coordinates. A smooth straight wire rotates in a plane with constant angular velocity about one end. Show that a particle which is free to slip along the wire may describe an equiangular spiral.
Prove that the linear momentum is conserved in a collision between two bodies. A body of mass \(m\) rests on a smooth table. Another of mass \(M\) moving with velocity \(V\) collides with it. Both are perfectly elastic and smooth and no rotations are set up by the collision. The body \(m\) is driven in a direction at angle \(\theta\) to the previous line of the body \(M\)'s motion. Show that its velocity is \(\frac{2M}{M+m}V \cos\theta\). Show further that if the subsequent motions of the two bodies are in perpendicular directions the masses must be equal.