SHM of a fly wing

Question

A fly beats its wings at a frequency of 1200 Hz. If the expansion and contraction of the wing muscles of the fly exhibits simple harmonic motion, the angular displacement of each wing of the fly also exhibits simple harmonic motion, the length of the wing muscle varies from 750 µm to 600 µm over the course of one beat of its 1.00 cm long wings, and the wing tips move through an arc of 150° then:

a) What is the time period of a single beat of the fly's wings?

b) What is the maximum velocity of the mobile end of the wing muscle?

c) What is the maximum angular velocity of the wing?

d) What is the maximum speed of the wing tip?

Answer 1

Period = T = 1/f = 1/1200= 8.33*E-4 seconds

Maximum velocity of muscle = Amplitude of movement * radian frequency = (75E-6)*2400pi = .565 m/sec

Amplitude = (Max-Min)/2 = (750-600)/2 = 75 micrometers (microns)

Radian frequency = 2pif = 2400pi

Maximum angular velocity of wing = Amplitude * radian frequency = 1.31*2400pi = 9869 rad/sec

Amplitude = 150/2 degrees = 75 degrees = 75*pi/180= 1.31 radians

Maximum speed of wingtip = Amp of movement * radian frequency = .0131*2400pi = 98.8 m/s

Arc length of wingtip motion = wing length * angle = .01 *150*pi/180= .0262 meters

Amplitude = arc length/2 = .0131 meters