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.
The angular points of a rectangle A, B, C, D are the middle points of the sides of a plane quadrilateral of which the lengths of two opposite sides are given. Construct the quadrilateral.
Construct the common tangents to two given circles. The radical axis of two circles external to each other intersects the external common tangents in H, H' and an internal common tangent in K, prove that KH.KH' is equal to the rectangle contained by the radii of the circles.
Prove that the polar reciprocal of one circle with respect to another is a conic, and find the position of its asymptotes and foci. P, P' are points on the auxiliary circle of an ellipse at the extremities of a diameter. The lines from P to the foci of the ellipse when produced meet the circle in T and T' respectively. Prove that P'T, P'T' are tangents to the ellipse.