If R' is the resultant when the angle is 2i 6. How many vibrations will a pendulum 39.3 inches long make in 10 minutes at a place where g=32.2? 7. A pendulum 93 inches long makes 120 vibrations in one minute at a certain place. Find the acceleration due to gravity at that place. The 8. A simple pendulum beats exactly seconds at a certain place, when the temperature is 0° C. pendulum wire is made of platinum which expands by 1 of its own length for a rise of temperature from 0° C. to 100° C. Find how much a clock regulated by this pendulum would lose per day at the same place when the temperature is 14° C. [Let l-length of pendulum at 0° C. Then the length (1+100). Thus time of vibration at 14° C. 9. A pendulum which beats seconds at a certain place where g=32.2 is carried to a place where g=32-197. How much would a clock regulated by this pendulum lose or gain per day? [It would lose 4 seconds per day.] 10. If W be the mass of the bob of a simple pendulum L the angle which the cord makes with the vertical when the pendulum is at the extremity of its range, the angle which the cord makes with the vertical in any other position of the pendulum, and T the tension of the cord, prove that T-Wg(3 cos 0-2 cos L). [Let O be the point of suspension of the pendulum. A the extremity of the range, and B the point where the cord makes an angle with the vertical. Let be the tension of the cord at the point B, if B were the extremity of its range. Then Wg cos 0. The velocity acquired by the bob in falling from A to B is found from V2=2gs=2gl(cos —cos L) and the centrifugal force at the point B is therefore =Wx2g(cos 0-cos L). The tension at the point B therefore T= = Wg cos 0+2 Wg(cos 0—cos L) = Wg(3cos 0 — 2cos L).] 11. A simple pendulum the mass of whose bob is 100 grammes swings through an arc of 5° on each side of the position of equilibrium. Find the tension of the cord (1) when the pendulum is at the extremity of its range; (2) when the pendulum is passing through the position of equilibrium. [(1) T=99-62 grammes weight; (2) T=100-76 grammes weight.] 12. The semivertical angle of a conical pendulum is 20° and the mass of the bob 2 kilogrammes. Find the tension of the cord. Find also the period and linear velocity of the bob, if the length of the cord is that of the second's pendulum. 9397 [To find T', the tension of the cord, by resolving vertically we have T'cos 0W; .. T=2·13 grammes weight. To find T the period, by resolving horizontally, we have T'sin 0 Ww2r; .. Ww2l=T= = Wg cos Ꮎ cos Ꮎ g =2cos 0.π =2 × 969-1938 secs. To find V, the linear velocity, we have SIMPLE HARMONIC MOTION. 13. Define Simple Harmonic Motion. A particle is moving with simple harmonic motion over a range of 1 feet on each side of the middle position five times per second. Find the maximum velocity and the velocity at one eighth of the period after leaving the middle position. [If at every instant a perpendicular AB be drawn from A, which moves in a circle with uniform velocity, to a fixed diameter of the circle, then the motion of B is simple harmonic. 14. Define the terms Amplitude, Epoch, Period, Phase, as used in simple harmonic motion. [See Thomson & Tait, § 71.] 15. Prove the formula x=R sint, which gives the Τ displacement x at any time t of a point executing simple harmonic motion, of amplitude R and period T. 16. A particle weighing lb. moves backward and forward in a straight line, 3 inches long, with simple harmonic motion 25 times per second. Find the force at the ends of the range, also the forces at a quarter and at one half the maximum distance from the centre. [Force at ends of range mw2r kinetic units. Force at a quarter distance from centre = {Mw2r =154 22 poundals. Force at half distance from centre ΙΟ =308 437 poundals.] 17. A mass of of a gramme executes 512 simple harmonic periods per second, and the amplitude of its motion is of a centimetre on each side of the middle position. Find the maximum force and the force at a distance of of a centimetre from the middle. 20 [258736-128 dynes=263.64 grammes weight. CENTRIFUGAL FORCES. 20 18. A mass of th gramme revolves in a circle one-third of a centimetre in radius and makes 65 revolutions per second. Find the centrifugal force in dynes and in grammes weight. (g=981 centimetres per second per second.) 19. Two particles each a pound mass revolve in circular orbits. The period of the one is 08 of a second, and the radius of its orbit 4 inches, the period of the other is 27 of a second, and the radius of its orbit 9 inches. Find the force acting on the body |