A uniform rigid rod \(AB\) of length 5 inches and weight \(w\) hangs from a point \(O\) by two inextensible strings \(AO\), \(BO\) of lengths respectively 3 and 4 inches. A variable weight \(W\) is attached at \(B\). Find the tension in \(OA\), and verify that it decreases as \(W\) increases.
A fisherman weighing 150 lb. gets into a boat and rows to the centre of a lake, where he drops anchor. After catching four fish of specific gravity 1, weighing 21 lb. altogether, he raises his anchor, rows to the bank, and goes home with his catch. The free surface of the lake has area 16,000 sq. yd., lake-water weighs 62½ lb. per cu. ft., and the anchor has weight 21 lb. and specific gravity 7. How does the level of the lake vary, in millionths of an inch?
The vertical cross-section of a smooth bowl is a parabola with equation \(r^2 = 4ah\), \(r\) being the radius at a height \(h\) above the bottom of the bowl. A needle (whose centre of gravity is at its mid-point) of length \(25a/4\) is put in the bowl. Discuss the possible positions of equilibrium of the needle.
Two cylinders lie in contact with axes horizontal on a plane inclined at 30° to the horizontal; the lower cylinder has radius \(r\) and mass \(m\) and the upper has radius \(3r\) and mass \(M\). Between the cylinders the coefficient of friction is \(\mu\), and between each cylinder and the plane the coefficient of friction is \(\mu'\). Show that the system is in equilibrium so long as \(3M > m\) and both \(\mu\) and \(\mu'\) exceed $$\sqrt{3(M + m)/(3M - m)}.$$
An intelligent fly can fly with speed \(u\) (relative to the air); it can also crawl with speed \(v\) directly into a wind, but not in any other direction. A wind is blowing with velocity \(V\) from the north. Distinguishing the cases \(u^2 \gtrless V(V + v)\), find how long the fly takes (by first flying and then crawling back north if necessary) to reach a point distant \(d\) due east.
A constant power \(P\) is available for turning a water-wheel of moment of inertia \(I\). A constant couple \(G\) opposing the rotation of the wheel is combined with a friction couple; find the time taken for the angular velocity to reach after \(\frac{P^2 I}{2G^3}(\log 2 - \frac{1}{2})\) revolutions.
Two equal light rods \(AB\), \(BC\) are freely jointed at \(B\) and lie on a smooth table. A heavy weight is attached at \(A\), and the point \(C\) is fixed. The rod \(BC\) is constrained to pass through a fixed point. Initially, \(A\) is at rest and \(ABC\) are in a straight line, and roughly describe the motion of \(A\).
Three equal heavy particles \(XYZ\) lie in a straight line on a smooth table. \(XY\) and \(YZ\) are joined by similar light springs, each of natural length \(L\). Initially, the particles are still, and the distance \(XY\) is \(L\), and the distance \(YZ\) is \(L - y_0\) (\(y_0 < L\)). Describe the subsequent motion—in particular, show that it is periodic.
Three beads \(ABC\) of equal mass are threaded in order on a smooth horizontal straight wire. The coefficient of restitution between the beads is \(e\), where \(0 < e < 1\). Initially, \(B\) and \(C\) are at rest, and \(A\) is moving towards \(B\). Show that there will always be at least three collisions. In particular, show that if $$1 - 6e + e^2 > 0$$ there will be at least five collisions.
An aeroplane flying with uniform velocity, not vertically, drops a bomb aimed accurately to hit a fixed gun. The gun fires a shell, aimed accurately to hit the aeroplane. [Aeroplane gun, bomb, and shell are all taken as ideal points, and air resistance is ignored.] You are warned that detailed calculation should be unnecessary.]