A carousel with a 50-foot diameter makes 4 revolutions per minute.
a) Find the angular speed of the carousel in radians per minute.
b) Find the linear speed in feet per minute of the platform rim of the carousel.
a) To find the angular speed of the carousel in radians per minute, we need to convert the number of revolutions per minute to radians per minute.
The conversion factor for going from revolutions to radians is 2π radians per revolution.
So, the angular speed in radians per minute can be calculated as follows:
Angular Speed (in radians per minute) = Number of Revolutions per Minute x 2π
Given that the carousel makes 4 revolutions per minute, we can plug in the value:
Angular Speed = 4 x 2π
Angular Speed = 8π radians per minute
b) The linear speed in feet per minute of the platform rim of the carousel can be calculated by using the formula:
Linear Speed = Angular Speed x Radius
Given that the diameter of the carousel is 50 feet, the radius is half of the diameter, which is 25 feet.
So, the linear speed can be calculated as follows:
Linear Speed = Angular Speed x Radius
Linear Speed = (8π radians per minute) x (25 feet)
Linear Speed = 200π feet per minute
Therefore, the linear speed of the platform rim of the carousel is 200π feet per minute.
a) To find the angular speed of the carousel in radians per minute, we need to convert the number of revolutions per minute to radians.
There are 2π radians in one revolution. So, the angular speed of the carousel in radians per minute is calculated as follows:
Angular speed (in radians per minute) = number of revolutions per minute * 2π radians/revolution
In this case, the carousel makes 4 revolutions per minute.
Angular speed = 4 revolutions per minute * 2π radians/revolution
Angular speed = 8π radians per minute
Therefore, the angular speed of the carousel is 8π radians per minute.
b) To find the linear speed in feet per minute of the platform rim of the carousel, we need to find the circumference of the carousel's platform by using the formula:
Circumference = π * Diameter
The diameter of the carousel is given as 50 feet. So, the circumference is:
Circumference = π * 50 feet
Circumference = 50π feet
Now, we know that the carousel makes 4 revolutions per minute. Each revolution covers the circumference of the platform rim. So, the linear speed of the platform rim is:
Linear speed (in feet per minute) = number of revolutions per minute * circumference
Linear speed = 4 revolutions per minute * 50π feet per revolution
Linear speed = 200π feet per minute
Therefore, the linear speed of the platform rim of the carousel is 200π feet per minute.
Answer
angular speed is frequency * 2pi = 8pi rad/min
linear speed = angular speed * radius = 8pi * 25 = 200pi ft/min