Final answer:
a. The period of rotation is 0.008 seconds. b. The frequency is 125 Hz. c. The angular speed (rotational) is 250π rad/s. d. The distance covered by a point on the outer edge of the CD is 0.16π meters. e. The centripetal acceleration experienced by the ant is (62500π²) m/s². f. The force responsible for keeping the ant on the disc is 37500π² mg N. g. The fastest angular and linear speed that the disc can spin before the ant will fall off is ω = √((0.072 mg (0.08))/(0.60 mg / 0.08)) and v = √((0.72 (g)r)/ 0.60).
Step-by-step explanation:
a. The period of rotation can be found using the formula T = 1/f, where T is the period and f is the frequency. Since the CD spins at 7500 revolutions per minute, which is equivalent to 125 revolutions per second, the frequency is 125 Hz. Therefore, the period of rotation is 1/125 = 0.008 seconds.
b. The frequency is 125 Hz, as calculated in part a.
c. The angular speed (rotational) can be found using the formula ω = 2πf, where ω is the angular speed and f is the frequency. Substituting the frequency of 125 Hz, we get ω = 2π(125) = 250π rad/s.
d. The distance covered by a point on the outer edge of the CD can be found using the formula d = 2πr, where d is the distance and r is the radius. The radius can be calculated by dividing the diameter by 2, so r = 160/2 = 80 mm = 0.08 m. Substituting this value into the formula, we get d = 2π(0.08) = 0.16π meters.
e. The centripetal acceleration experienced by the ant can be found using the formula ac = ω²r, where ac is the centripetal acceleration, ω is the angular speed, and r is the radius. Substituting the given values, we get ac = (250π)²(0.08) = (62500π²) m/s².
f. The force responsible for keeping the ant on the disc is the centripetal force, which can be calculated using the formula Fc = mac, where Fc is the centripetal force, m is the mass of the ant, and ac is the centripetal acceleration. Substituting the given values, we get Fc = (0.60 mg)(62500π²) = 37500π² mg N.
g. To find the fastest angular and linear speed that the disc can spin before the ant will fall off, we need to consider the maximum frictional force that can be exerted. The maximum frictional force is given by Fmax = μN, where μ is the coefficient of friction and N is the normal force. The normal force can be calculated using the formula N = mg, where m is the mass of the ant and g is the acceleration due to gravity. Substituting the given values, we get N = (0.60 mg) N. Therefore, the maximum frictional force is Fmax = (0.12)(0.60 mg) = 0.072 mg N. Since the frictional force provides the centripetal force, we can equate the maximum frictional force to the centripetal force to find the maximum angular and linear speeds. Using the formula Fmax = mac and the known values of Fmax and ac, we can solve for ω, which represents the angular speed, and v, which represents the linear speed. The equation becomes Fmax = mω²r = mv²/r, where r is the radius of the CD. Simplifying this equation, we get: ω²r = v², and substituting the known values, 0.072 mg (0.08) = (0.60 mg/ 0.08) v²
The angular speed can therefore be calculated as ω = √((0.072 mg (0.08))/(0.60 mg / 0.08))
The linear speed can be calculated as v = ωr = √((0.72 (g)r)/ 0.60)