towards the centre of the circle in each case, and the values in the two cases. compare 2 4 [F=1 × (27) × 11 × 51-5 pounds weight 08 12 20. Prove that if two equal planets move in circles. round the same centre obeying Kepler's third law, the forces upon them towards the centre are inversely proportional to the squares of the radii of their circles. [Let F and F be the forces acting towards the centre on the two planets. Then 21. A ball weighing two kilogrammes is whirled in a vertical circle of 30 cms. radius, with such a speed as to keep the string stretched at the highest point and no more. Find (1) the velocity of the ball at the highest point; (2) the velocity of the ball at the lowest point; (3) the tension of the string at the latter point. [Let Th, V be tension and velocity at highest point: T, V, tension and velocity at lowest point. Then h = h =171.6 cms. per second. V=V2+ square of velocity acquired by falling through a height equal to diameter of circle 22. An indiarubber band whose mass is 3 grammes per 100 centimetres is stretched on the circumference of a wheel of 8 centimetres radius and has a tension of 20,000 dynes. The wheel being set in motion, find how many revolutions per second it must make in order that the band may not press upon it. [If T is tension of rope, 2T sin 10 or T0=centripetal force, ATWOOD'S MACHINE. 23. Describe and explain the mode of using Atwood's machine. The mass of each box being 26m, the inertia-equivalent of the wheel work 11m, find the velocity acquired and the space described from rest in 5 seconds when an additional mass 2m is placed on one of the boxes. v=at=84×5=41% feet per second; 24. The following experiment is made to determine the inertia-equivalent of the wheels of an Atwood's machine. Two equal weights each 14 ounces are attached to the ends of a fine cord passing over the main wheel. To one of these one ounce is added and the system is allowed to start from rest. In five seconds the falling weight is found to have moved through 12.5 feet. Calculate the inertia-equivalent. [Let I inertia-equivalent in ounces. Then 25. Prove that the attractions of homogeneous spheres of different densities on a particle placed at the same distance from the centre of each are as the products of the cubes of the radii and the densities of the spheres. where c is a constant, r its radius, p its density, and d the distance of the attracted particle from its centre. Similarly for second sphere 26. A small particle is placed at a distance of 3 feet from the centre of a ball of lead 1 ft. 6 inches in radius: find through what distance a particle will approach the centre of the ball in 20 seconds. Data. Radius of the earth 21 × 106 feet; density of lead 11:36; mean density of the earth 5.68. [For the earth we have 32_c$7(21 x 1083 × 568 C: 96 4 x T x 21 x 106 × 5.68° For globe of lead 32 27. A particle is first brought close to an iron ball of one foot radius, then close to a ball of lead of B two feet radius. Taking the density of iron as 7.8 and the density of lead as 114, compare the attractions experienced by the ball in the two cases. [For iron ball 28. A gyrostat is set with its axis horizontal on a smooth glass plate. A mass of 80 grammes hung on the cover of the axle at a distance of six centimetres from the centre of inertia of the gyrostat, causes the axis to turn in azimuth with a velocity of of a radian per second. The mass of the wheel is 1800 grammes and its radius of gyration 4 cms. Find the angular velocity of the rotating wheel. |