A 65.0 cm diameter wheel accelerates uniformly from 230 rpm to 360 rpm in 5.0 s. How far will a point on the edge of the wheel have traveled in this time?

Sure, here's how to solve this problem:

Step 1: Convert the diameters to radius and the radius to meters. The radius is half of the diameter, so the radius of the wheel is 65.0 cm / 2 = 32.5 cm. Convert this to meters by dividing by 100, so the radius is 0.325 m.

Step 2: Convert the rotations per minute (rpm) to radians per second. There are 2π radians in one rotation and 60 seconds in one minute. So, 230 rpm = 230 * 2π / 60 = 24.084 rad/s and 360 rpm = 360 * 2π / 60 = 37.699 rad/s.

Step 3: Calculate the angular acceleration. The formula for angular acceleration is (ωf - ωi) / t, where ωf is the final angular speed, ωi is the initial angular speed, and t is the time. So, the angular acceleration is (37.699 rad/s - 24.084 rad/s) / 5.0 s = 2.723 rad/s².

Step 4: Calculate the total angle the wheel has turned through. The formula for the total angle turned through during a period of uniform acceleration is ωit + 0.5α*t², where α is the angular acceleration. So, the total angle turned through is 24.084 rad/s * 5.0 s + 0.5 * 2.723 rad/s² * (5.0 s)² = 120.42 rad + 34.0375 rad = 154.4575 rad.

Step 5: Calculate the distance traveled. The distance traveled by a point on the edge of the wheel is the total angle turned through times the radius of the wheel. So, the distance traveled is 154.4575 rad * 0.325 m = 50.1989375 m.

So, a point on the edge of the wheel will have traveled approximately 50.2 meters in this time.