If an 22-inch tire rotates 40 times per 2 minute, determine th elinear and angular speed of the tire.
To determine the linear and angular speed of the tire, we need to understand a few concepts:
1. Circumference of a tire: The circumference of a circle is given by the formula C = 2πr, where r is the radius of the tire.
2. Angular speed: Angular speed, also known as rotational speed, is the rate at which an object rotates around a fixed axis. It is usually measured in radians per second (rad/s).
3. Linear speed: Linear speed is the distance traveled by an object along a circular path per unit of time. It is usually measured in meters per second (m/s).
Now, let's solve the problem step by step:
Step 1: Convert the tire size from inches to meters.
Since the tire size is given in inches, we need to convert it to meters to maintain consistent units. We know that 1 inch is equal to 0.0254 meters. Therefore, a 22-inch tire is equal to 22 * 0.0254 = 0.5588 meters.
Step 2: Calculate the circumference of the tire.
The circumference of a circle is given by C = 2πr, where r is the radius of the tire. In this case, since we have the diameter (22 inches), we need to divide it by 2 to get the radius.
Radius = Diameter / 2 = 22 inches / 2 = 11 inches.
Convert the radius to meters: 11 inches * 0.0254 = 0.2794 meters.
Now, we can calculate the circumference: C = 2πr = 2π * 0.2794 meters.
Step 3: Calculate the linear speed.
The linear speed of an object can be calculated using the formula: v = s / t, where v is the linear speed, s is the distance traveled, and t is the time taken.
We are given that the tire rotates 40 times in 2 minutes. In each rotation, the distance traveled is equal to the circumference of the tire. Therefore, the total distance traveled in 40 rotations is 40 * C.
Now, divide the distance by the time to get the linear speed: v = (40 * C) / 2 minutes.
Step 4: Calculate the angular speed.
Angular speed (ω) is defined as the angle turned in a given time. In this case, the tire makes one full rotation (360 degrees) in one revolution.
The angular speed can be calculated using the formula: ω = θ / t, where ω is the angular speed, θ is the angle turned, and t is the time taken.
Since the tire makes one full rotation in each revolution, the angle turned is 360 degrees or 2π radians. The time taken is given as 2 minutes.
Substitute the values into the formula: ω = (2π) / 2 minutes.
Step 5: Simplify the equations.
Simplify the linear speed by substituting the previously calculated circumference: v = (40 * C) / 2 minutes.
Similarly, simplify the angular speed: ω = (2π) / 2 minutes.
Step 6: Convert the time units to seconds.
To convert the time from minutes to seconds, multiply by 60 seconds per minute.
By following these steps, we can calculate both the linear and angular speed of the 22-inch tire based on the given information.