A ceiling fan has

15
-inch blades (so the radius of the circular fan is
15
inches). Suppose the fan turns at a rate of
60
revolutions per minute.
(a) Find the angular speed of the fan in radians per minute.
(b) Find the linear speed of the tip of a blade in feet per second.

Do not round any intermediate computations, and round your answer to the nearest whole number.

A ceiling fan has

15
-inch blades (so the radius of the circular fan is
15
inches). Suppose the fan turns at a rate of
60
revolutions per minute.
(a) Find the angular speed of the fan in radians per minute.
(b) Find the linear speed of the tip of a blade in feet per second.

(a) To find the angular speed of the fan in radians per minute, we need to convert the number of revolutions per minute to radians per minute.

Since each revolution is equivalent to
2๐œ‹ radians, we can multiply the number of revolutions per minute by 2๐œ‹ to get the angular speed in radians per minute.

Angular speed = 60 revolutions per minute * 2๐œ‹ radians per revolution = 120๐œ‹ radians per minute.

(b) To find the linear speed of the tip of a blade in feet per second, we need to find the circumference of the circular path traced by the tip of the blade.

The circumference is given by the formula
circumference = 2๐œ‹ * radius.

So, the circumference of the fan is 2๐œ‹ * 15 inches.

To convert inches to feet, we divide by 12:
circumference = (2๐œ‹ * 15 inches) / 12 = ๐œ‹ * 15/6 feet.

Since the fan makes one revolution in one minute, the linear speed of the tip of the blade is equal to the circumference of the fan multiplied by the number of revolutions per minute.

Linear speed = circumference * revolutions per minute = (๐œ‹ * 15/6 feet) * 60 revolutions per minute.

To round the answer to the nearest whole number, we can calculate the value.

Linear speed = (3.1416 * 15/6 feet) * 60 revolutions per minute โ‰ˆ 235 feet per minute.

Converting minutes to seconds, we divide by 60:
235 feet per minute / 60 = 3.92 feet per second.

Rounding to the nearest whole number, the linear speed of the tip of the blade is approximately 4 feet per second.

To find the answer to part (a), we need to convert the revolutions per minute to radians per minute.

1 revolution is equal to 2ฯ€ radians.

Therefore, the angular speed in radians per minute is calculated as:

Angular Speed (in radians per minute) = 60 revolutions per minute * 2ฯ€ radians per revolution

Angular Speed = 120ฯ€ radians per minute

To find the answer to part (b), we need to find the linear speed of the tip of the blade.

The linear speed is the distance traveled by the tip of the blade in a certain time.

To calculate the linear speed, we need to know the circumference of the circle traced by the tip of the blade. The circumference can be found using the formula:

Circumference = 2ฯ€ * radius

Given that the radius of the fan's blade is 15 inches, the circumference is:

Circumference = 2ฯ€ * 15 inches

Now, we need to convert the linear speed from inches per minute to feet per second.

There are 12 inches in 1 foot and 60 seconds in 1 minute.

Therefore, the linear speed in feet per second is calculated as:

Linear Speed (in feet per second) = (Circumference in inches / 12 inches per foot) * (1 foot / 60 seconds)

Linear Speed = (2ฯ€ * 15 inches / 12 inches per foot) * (1 foot / 60 seconds)

Linear Speed = ฯ€ / 2 feet per second

Finally, we can round the answer to the nearest whole number, which gives us:

Linear Speed โ‰ˆ 2 feet per second

60 rev/min * 2ฯ€ rad/rev = 120ฯ€ rad/min

the linear speed is just radius * angular speed

120ฯ€/min * 15in * 1ft/12in * 1min/60s = (120ฯ€*15)/(12*60) ft/s