A Ferris Wheel rotates 3 times each minute. The passengers sit in seats that are 25 feet from the center of the wheel.
A. What is the angular velocity of the wheel in degrees per minute and radians per minute?
B. What is the linear velocity of the passenger in the seats?
A. Angular velocity is defined as the change in angle with respect to time. Since the Ferris Wheel completes 3 rotations per minute, we can calculate the angular velocity.
To find the angular velocity in degrees per minute, we multiply the number of rotations by 360 (since there are 360 degrees in a full circle).
Angular velocity (in degrees per minute) = 3 rotations/min * 360 degrees/rotation = 1080 degrees/min
To convert this to radians per minute, we use the conversion factor: 1 radian = π/180 degrees.
Angular velocity (in radians per minute) = 1080 degrees/min * π/180 degrees/radian = 18π radians/min
B. Linear velocity is the distance traveled per unit of time. In this case, we need to find the linear velocity of the passenger in the seats.
The distance traveled by the passenger is equal to the circumference of the circle formed by the seats. The formula for circumference is given by:
Circumference = 2πr
where r is the distance from the center of the wheel to the seats (25 feet).
Circumference = 2π * 25 feet = 50π feet
Since the Ferris Wheel completes 3 rotations per minute, the linear velocity can be calculated by dividing the circumference by the time taken for 3 rotations.
Linear velocity = (50π feet)/(3 rotations) = (50π feet/rotation) / (3 rotations/min) = 50/3 π feet/min (approximately)
So, the linear velocity of the passenger in the seats is approximately 16.67π feet per minute.
To find the angular velocity of the Ferris Wheel, we need to determine the angle it rotates through in a given time period.
A. Angular velocity is given by the formula:
angular velocity = (number of revolutions) x (360 degrees) / (time in minutes)
In this case, the Ferris Wheel rotates 3 times in 1 minute.
Angular velocity = 3 x 360 degrees / 1 minute
= 1080 degrees per minute
To convert this value to radians per minute, we use the fact that 1 revolution is equal to 2π radians:
angular velocity in radians per minute = 1080 degrees per minute x (2π radians / 360 degrees)
= 6π radians per minute
Therefore, the angular velocity of the Ferris Wheel is 1080 degrees per minute and 6π radians per minute.
B. The linear velocity of a point on the Ferris Wheel is given by the formula:
linear velocity = angular velocity x radius
In this case, the passengers sit at a distance of 25 feet from the center of the wheel.
linear velocity = angular velocity x radius
= (6π radians per minute) x (25 feet)
= 150π feet per minute
Therefore, the linear velocity of the passengers in the seats is 150π feet per minute.
471.25 feet/min
3 rpm = 6π radians/min
2π*25*3 = 150π ft/min