Question

A cyclist starts from rest and pedals so that the wheels make 8.0 revolutions in the first \(5.0 \mathrm{s}\). What is the angular acceleration of the wheels (assumed constant)?

Short answer

Answer: The angular acceleration of the bicycle wheels is 1.60 rad/s².

Step-by-step solution

Convert revolutions to radians

Since 1 revolution is equal to \(2\pi\) radians, we can convert the number of revolutions to radians using this relationship. The wheels make 8.0 revolutions, so the total angular displacement is \(8.0 \cdot 2\pi\) radians.

Calculate average angular velocity

The average angular velocity can be calculated by dividing the total angular displacement by the time interval. The cyclist pedals for 5.0 seconds, so the average angular velocity is \(\omega_{avg} = \frac{8.0 \cdot 2\pi}{5.0}\).

Write down the equation of motion for angular acceleration

Now we can use the equation of motion to find the angular acceleration \(\alpha\). The equation of motion that relates angular displacement, initial angular velocity, angular acceleration, and time is: $$\theta = \omega_{0}t + \frac{1}{2} \alpha t^2$$ In this case, the cyclist starts from rest, so the initial angular velocity \(\omega_{0}\) is 0. The angular displacement \(\theta\) is \(8.0 \cdot 2\pi\), and the time t is 5.0 s. We can rewrite the equation as: $$8.0 \cdot 2\pi = 0 \cdot 5.0 + \frac{1}{2} \alpha (5.0)^2$$

Solve for angular acceleration

Solving the equation for angular acceleration (\(\alpha\)), we have: $$\alpha = \frac{2 \cdot (8.0 \cdot 2\pi)}{(5.0)^2}$$ Calculate the value of angular acceleration: $$\alpha = \frac{2 \cdot (8.0 \cdot 2\pi)}{(5.0)^2} = 1.60 \ rad/s^2$$ The angular acceleration of the bicycle wheels is \(1.60 \ rad/s^2\).