Question

A cyclist accelerates from rest at a rate of 1.00 m/s^2. How fast will a point at the top of the rim of the tire (diameter 68.0 cm) be moving after 2.25 s?

Answer 1

The point at the top of the rim of the tire will be moving at a speed of approximately 0.56 m/s after 2.25 seconds of acceleration from rest at a rate of 1.00 m/s².

Step-by-step explanation:

To calculate the speed of the point at the top of the rim of the tire, we need to find the angular velocity at that point first. The cyclist accelerates at a rate of 1.00 m/s², which corresponds to angular acceleration. Using the formula α = a/r, where α is the angular acceleration, a is the linear acceleration, and r is the radius of the tire, we can calculate the angular acceleration as 1.00 m/s² divided by half the diameter of the tire (0.34 m). This gives us an angular acceleration of approximately 1.47 rad/s².

To find the angular velocity at the top of the rim, we can use the formula ω = αt, where ω is the angular velocity and t is the time. Plugging in the values, 1.47 rad/s² and 2.25 s, we can find that the angular velocity is approximately 3.31 rad/s.

Finally, to convert the angular velocity to linear velocity, we can use the formula v = ωr, where v is the linear velocity and r is the radius of the tire. Plugging in the values, 3.31 rad/s and 0.17 m (half the diameter of the tire), we can find that the point at the top of the rim of the tire will be moving at a speed of approximately 0.56 m/s².

Answer 2

The speed of the top of the rim after 2.25 s is 4.50 m/s, given that the bicycle accelerates from rest at 1.00 m/s². This is because the topmost point on the rim moves with twice the bicycle's linear speed due to rotational motion in addition to the bike's translation.

Step-by-step explanation:

The speed of a point at the top of the rim of a bicycle tire after 2.25 seconds of accelerating from rest at a rate of 1.00 m/s2 can be found by considering the tangential speed of the rim. The tangential speed is the same as the linear speed of the bike because one point on the rim rotates around the axle as the bike moves forward. As the bike accelerates linearly, so does every point on the rim.

First, calculate the linear velocity (v) using the formula v = u + at, where u is the initial velocity, a is the acceleration, and t is the time. Substituting the given values, we get v = 0 + (1.00 m/s2)(2.25 s) = 2.25 m/s.

Now, remember that the topmost point on a wheel traveling linearly will be moving at a speed equal to twice the linear speed of the bike because it combines rotational and translational motion. Therefore, this point will be moving at 2 × 2.25 m/s = 4.50 m/s.