Two ladders are connected as shown in the figure. The rungs at B are lashed together and the end C of the ladder AC touches the ladder BD. (Friction at D and C may be neglected.) Show that the pull on the lashing due to a man of weight W on the rung E is \[ \frac{Wc \sin \theta c^{-2}}{a+b+c\left(b^2/c^2+\tan^2\theta\right)^{1/2}}. \] \emph{[A diagram shows two ladders, AC and BD. BD is leaning against a vertical wall and horizontal floor, forming an angle \(\theta\) with the wall. AC is attached to BD. A point E is on AC, and a weight W hangs vertically from E. The segments from A to E, E to C, C to B are labeled c, b, a respectively.]}
Find the centre of mass of a uniform solid hemisphere. If the hemisphere is suspended by a string passing through a smooth fixed ring and attached to two points on the rim of the plane face at opposite ends of a diameter, show that in equilibrium with the plane face inclined to the horizontal, \(8 \tan\theta = 3\), where \(2\theta\) is the angle between the parts of the string.
The distance between the axles of a railway truck is \(a\), and the centre of gravity is halfway between them and at a perpendicular distance \(h\) from the rails. With the lower wheels locked the greatest incline on which the truck can rest is \(\alpha\). Show that the coefficient of friction between the wheels and rails is \[ 2a/(a\cot\alpha+2h). \]
A projectile is fired from a point O with velocity due to a fall of 100 feet from rest and hits a mark 50 feet below O and horizontally distant 100 feet from O. Show that the two possible directions of projection are at right angles, and calculate their inclination to the horizontal.
The bar AC hinged to the wall at A is supported horizontally by a chain of rods attached to B. The lengths of the vertical tie-rods are so arranged that the tensions in them are equal and equal to the vertical pull of the chain at C. Determine the lengths of the tie-rods in terms of \(a\). \emph{[A diagram shows a horizontal bar AC, hinged at A to a vertical wall. A point B is on the wall above A. A chain of rods connects B to C. The chain consists of several links connected by vertical tie-rods to the bar AC. The horizontal distance between the tie-rods is \(a\).]}
The resistance to an aeroplane when landing is \(a+bv^2\) per unit mass, \(v\) being the velocity, \(a, b\) constants. For a particular machine, \(b=10^{-2}\) ft-lb.-sec. units and it is found that if the landing speed is 50 miles per hour the length of run of the machine before coming to rest is 150 yards. Calculate the value of the constant \(a\).
A ship is making \(n\) complete rolls a minute and the motion of the masthead \(h\) feet above sea level may be taken as a horizontal simple harmonic motion of total extent \(2a\). When at a distance \(x\) from the mean position a weight falls from the masthead. Find where it will hit the water, and prove that the distance of this point from the ship will be a maximum when \[ x=a\left(1+\frac{hn^2}{1470}\right)^{-\frac{1}{2}}, \text{ approx.} \]
Show that the acceleration of a particle along the normal to its path is \(v^2/\rho\), where \(\rho\) is the radius of curvature and \(v\) the velocity. A bead moves on a smooth parabolic wire whose axis is vertical and vertex upwards. Show that the pressure between the wire and bead varies inversely as \(\rho\).
A gun of mass \(M\) fires a shell of mass \(m\) horizontally, and the energy of the explosion is such as would be sufficient to project the shell vertically to a height \(h\). Show that the velocity of recoil of the gun is \[ \{2m^2gh/M(M+m)\}^{\frac{1}{2}}. \]
Two unequal masses are connected by a string of length \(l\) which passes through a fixed smooth ring. The smaller mass moves as a conical pendulum while the other mass hangs vertically. Find the semi-angle of the cone, and the number of revolutions per second when a length \(a\) of the string is hanging vertically.