Show that, if a point moves along any curve under the action of a force always at right angles to the direction of motion, the point moves with constant speed. A particle is attached to the end of a string which is partly wound round a post whose section is a regular polygon of \(r\) sides each of length \(a\). Initially a length \(l\), equal to an integral multiple of \(a\), is unwound and in a straight line with one of the sides. The particle is then projected at right angles to the string with velocity \(v\) so that the string winds in a horizontal plane round the post. Show that the time taken to wind up is \[ \frac{\pi l(l+a)}{rav}. \]
An aqueduct of cross section 2 sq. ft. delivers water with a velocity of 2 ft. per sec. at the top of a water wheel of 12 ft. diameter. The water leaves the wheel at a point 3 ft. below the centre. Calculate the horse-power developed. The compartments which catch the water being 40 in number, find the minimum capacity of each in order that all the water may be caught when the rim velocity of the wheel does not exceed that with which the water is delivered to it; calculate also the maximum turning couple which the wheel can exert when working at full power; examine the effect of friction at the axle, and discuss what happens if the load is greater than the couple calculated above.
The figure represents a freely jointed framework supporting the wings of an aeroplane. The load is represented by a force \(W\) at the middle point of the lower side of the framework and the upthrust of the air by equal forces \(P ( = W/10)\) at the joints. Find graphically the stresses in the various members, and show which may be wires, and which should be rigid bars. [The figure shows a rectangular truss framework, 6' high, composed of four 8' by 6' panels, with diagonal bracing. A downward force W acts at the central lower joint. Upward forces P act at all ten joints on the upper and lower chords.]
If three forces acting on a body are in equilibrium, show that they are coplanar and either concurrent or parallel. A smooth sphere of radius \(a\) and weight \(w\) is suspended by a string of length \(l\) from a hook. From the same hook a weight \(w'\) is hung by a long string. Shew that the angle which the first string makes with the vertical is \(\sin^{-1} \frac{w'a}{(w+w')(a+l)}\).
From the points B, C, D of a light string ABCDE weights proportional to 4, 8 and 5 are hung respectively. It is found that the portions of the string BC and CD make angles of 25\(^\circ\) and 15\(^\circ\) respectively with the horizontal. Find graphically the angles which the strings AB and DE make with the vertical.
A triangular frame formed of three uniform rods, jointed together at their extremities, of length 3, 4 and 5 ft respectively, is suspended by a string attached to the middle point of the longest side. Shew that the reactions at the joints are \[ W\sqrt{\frac{137}{8}}, \quad W\sqrt{\frac{109}{8}}, \quad W\frac{5}{2\sqrt{2}}, \] where \(W\) is the weight per foot of rod.
Out of a circular disc of metal a circle is punched whose diameter is a radius \(OA\) of the disc. The disc is then placed vertically resting on two rough parallel rails in the same horizontal plane, the plane of the disc being perpendicular to the rails. The chord of contact subtends an angle \(2\alpha\) at the centre of the disc. Shew that if the angle which \(OA\) makes with the vertical is greater than \(\sin^{-1}\left(\frac{3\sin 2\epsilon}{\cos\alpha}\right)\) where \(\epsilon\) is the angle of friction, the disc will slip.
In rectilinear motion, when the acceleration at consecutive intervals of time is given, shew how the velocity can be calculated by plotting the acceleration-time curve. A motor car is running at 10 M.P.H. when it starts to accelerate. The acceleration diminishes uniformly with the time and after 20 seconds the acceleration is zero and the car is running at 25 M.P.H. Sketch the velocity-time curve and calculate the distance travelled during the period of acceleration and the time that elapses before the speed of 20 M.P.H. is reached.
Two weights \(A\) and \(B\) are connected by a string passing over a smooth light pulley. To the weight \(B\) is attached another weight \(C\) by a string of length 2 ft. \(B\) and \(C\) are held initially in contact and resting on a platform vertically below the pulley. If the masses of \(A, B\) and \(C\) are 5, 3 and 4 lb. respectively, shew that when the system is free to move, the weight \(C\) will strike the platform again after \(12/\sqrt{g}\) seconds and that the weight \(B\) will come momentarily to rest at a distance \(1\frac{1}{2}\) ft from the platform.
A train of mass 200 tons is ascending an incline of 1 in 100, the resistance to the motion being 15 lb. wt per ton. When its velocity has reached 12 M.P.H., what is its acceleration if the H.P. then developed by the engine is 600?