Prove the law of conservation of energy for a particle performing simple harmonic motion. Hence graphically show the variation of kinetic energy and potential energy w.r.t. instantaneous displacement.
Two sound notes have wavelengths `(83)/(170)m and (83)/(172)m` in the air. These notes when sounded together produce 8 beats per second. Calculate the velocity of sound in the air and frequencies of the two notes.

Explanation

Data ` lamda_(1) = 83/170 m . Lamda_(2) = 83/172 m` beat frequency with each other = 8 Hz
Let `n_(1) and n_(2) ` be the corresponding frequencies and v the speed of sound in air.
`v = n_(1) lamda_(2) = n_(2)lamda_(2) " " "But" lamda_(1) gt lamda_(2) " " therefore n_(1) lt n_(2)`
` n_(2) - n_(1) = 8 Hz " " v/(lamda_(2)) - v/(lamda_(1)) = 8`
` v [ 172/83 -170/83] =8 " " therefore v xx 2/83 = 8`
` v = 4 xx 83 = 332 ` m/s
` n_(1) = v/lamda_(1) = 332 xx 170 /83 4 xx 170 = 680 ` Hz
` n_(2) = n_(1) + 8 = 680 + 8 = 688 Hz`