The countershaft of a lathe carries two gear-wheels whose pitch diameters are 6" and 3" respectively: the central planes of the wheels are 6" apart and the ends of the shafts run in bearings outside the wheels whose central planes are 2" from the central plane of the corresponding wheel. The gear-wheels on the mandrel have pitch diameters of 3" and 6", so that the total reduction ratio is 4:1, and the driving belt exerts a couple of magnitude 20 lb. ft. on the first of these two latter wheels. If the angle of obliquity of thrust between the gear teeth be 20\(^\circ\), calculate the load on each bearing of the countershaft, neglecting friction throughout.
State and prove any graphical construction for finding the acceleration of a steam-engine piston at any point of the stroke. Find the acceleration of the middle point of the connecting rod of an engine running at 400 r.p.m. at the instant when the crank is at right angles to the line of stroke, if the lengths of crank and connecting rod be 6" and 2' 0" respectively.
A circular galvanometer coil has a rectangular cross-section, the external and internal radii being 2 cm. and 1.5 cm. respectively, whilst the axial length is 1 cm. The coil is hung up by a bifilar suspension consisting of two light parallel strips each 15 cm. long, the strips being 0.5 cm. apart. Calculate the time of a small oscillation about a vertical basis midway between the strips.
A variable speed friction gear consists of a flat disc on a shaft running at uniform speed, in contact with the face of which runs a friction wheel on a shaft which intersects the first shaft at right angles, the wheel being pressed on to the disc with a pressure just sufficient to prevent slipping as a whole. Shew that the efficiency of the transmission is \(1/\left(1+\frac{a}{4r}\right)\), where \(a\) is the length of the line of contact of the friction wheel and \(r\) is the radius from the centre of the disc to the centre of that line of contact.
A uniform rod 8' long standing vertically on the ground falls over so that its centre strikes a horizontal bar: the bar is perpendicular to the length of the rod at the moment of impact and is 2' above the ground. Find the velocity with which the upper end of the rod strikes the ground, assuming that neither rod nor bar is bent by the impact.
Describe the type of fracture to be expected when round bars of good mild steel and good grey cast iron are broken (a) in tension and (b) in torsion. If the bars are 1" in diameter, suggest suitable values of the breaking load for each material in the two cases. Distinguish between "hardening" and "tempering" of high carbon steel. How is mild steel case-hardened?
The moment of inertia of cross-section of a cantilever of length \(l\) varies from \(I\) at the support to \(\frac{1}{4}I\) at the outer end, where a load \(W\) is carried. Shew that the deflection at the outer end is \(\frac{3Wl^3}{8EI}\log_e 3\).
The rotating parts of a motor-car engine may be considered as equivalent to a flywheel weighing 100 lb. and having a radius of gyration of 4\(\frac{1}{2}\)". When the engine is running at 1000 r.p.m. the clutch is accidentally let in suddenly with the car at rest and the gear engaged. If the clutch slips for \(\frac{1}{10}\) sec., after which the engine speed is 200 r.p.m., find the maximum stress reached in the clutch-shaft, which is tubular, with external and internal diameters of 1" and \(\frac{3}{4}\)" respectively.
On a thick cylinder, whose external and internal diameters are 6" and 4" respectively, is wound one layer of steel wire under a uniform tension during winding of 1000 lb., the diameter of the wire being 0.1". Calculate the greatest intensity of "hoop" compressive stress in the tube.