An ant sits on a CD 17 cm from the center. If it remains there for 42 seconds, it travels a total of 913 cm. a) How many rotations did the ant complete? b) At what speed was the ant traveling? (in cm/s) c) What is the period of the ant's revolutions? (in s) d) At what rotational speed is the CD turning? (in rpm)

To solve the problems, we need to consider the information given and apply formulas for circular motion.

a) To find the number of rotations the ant completed, we first need to calculate the circumference of the circle, which is the distance traveled in one complete rotation. The circumference (C) of a circle is given by the formula:

C = 2 * π * r

where r is the radius of the circle. Since the ant sits 17 cm from the center of the CD, the radius (r) is 17 cm.

C = 2 * π * 17 cm

C ≈ 2 * 3.14 * 17 cm

C ≈ 106.76 cm

Now, to find the number of rotations (N), we divide the total distance traveled by the ant (913 cm) by the circumference of the circle.

N = Total distance / Circumference

N = 913 cm / 106.76 cm

N ≈ 8.55 rotations

So, the ant completed approximately 8.55 rotations.

b) To calculate the speed (v) at which the ant was traveling, we use the formula:

v = Total distance / Total time

We were given the total distance (913 cm) and the total time (42 s), so:

v = 913 cm / 42 s

v ≈ 21.74 cm/s

The ant traveled at approximately 21.74 cm/s.

c) The period (T) of the ant's revolutions is the time it takes to complete one rotation. Since we know the number of rotations and the total time, we can calculate the period as follows:

T = Total time / Number of rotations

T = 42 s / 8.55 rotations

T ≈ 4.91 s

Therefore, it takes approximately 4.91 seconds to complete one rotation.

d) Rotational speed (ω) in revolutions per minute (rpm) can be calculated by knowing the number of rotations and the total time in minutes. First, we need to convert the total time from seconds to minutes:

Total time in minutes = Total time in seconds / 60 seconds

Total time in minutes = 42 s / 60 s/min

Total time in minutes ≈ 0.7 min

Now, to find ω in rpm:

ω = Number of rotations / Total time in minutes

ω = 8.55 rotations / 0.7 min

ω ≈ 12.21 rpm

So, the CD is turning at approximately 12.21 rotations per minute.