Question

You are holding the axle of a bicycle wheel with radius \(35.0 \mathrm{~cm}\) and mass \(1.00 \mathrm{~kg}\). You get the wheel spinning at a rate of 75.0 rpm and then stop it by pressing the tire against the pavement. You notice that it takes \(1.20 \mathrm{~s}\) for the wheel to come to a complete stop. What is the angular acceleration of the wheel?

Short answer

Answer: The angular acceleration of the bicycle wheel as it comes to a stop is -6.545 rad/s².

Step-by-step solution

Convert given angular velocity to rad/s

First, we need to convert the given angular velocity from rpm to radians per second. Recall that 1 revolution is equal to \(2\pi\) radians and 1 minute is equal to 60 seconds. Given: Angular velocity in rpm = 75.0 rpm So, \(\omega_i\) in radians/s: \(\omega_i = (75.0\ \text{rpm}) \times \frac{2\pi\ \text{radians}}{1\ \text{revolution}} \times \frac{1\ \text{minute}}{60\ \text{seconds}} = 7.854 \mathrm{~rad/s}\)

Use the angular kinematic formula to find the angular acceleration

Now, we can use the angular kinematic formula to find the angular acceleration: \(\omega_f = \omega_i + \alpha t\) We know that at the end of the process, the wheel comes to a complete stop. So, \(\omega_f = 0\) Plugging in the values: \(0 = 7.854 \mathrm{~rad/s} + \alpha (1.20 \mathrm{~s})\)

Solve for the angular acceleration

Isolating \(\alpha\): \(\alpha = - \frac{7.854 \mathrm{~rad/s}}{1.20 \mathrm{~s}}\) \(\alpha = - 6.545 \mathrm{~rad/s^2}\) The negative sign indicates that the angular acceleration acts to oppose the initial angular velocity, as the wheel comes to a stop. Therefore, the angular acceleration of the wheel is \(- 6.545 \mathrm{~rad/s^2}\).

Key concepts

Angular Velocity

Angular velocity is a vector quantity that represents the rate of change of an object's angular position over time. It essentially describes how fast an object is spinning or rotating. The standard unit for angular velocity is radians per second (\text{rad/s}). This measurement is more informative than revolutions per minute (rpm) as it gives a clearer picture of the angle covered by the object per unit of time, regardless of the size of the circle.

For example, if a bicycle wheel spins at 75.0 rpm, when converted to rad/s, we obtain a more standardised measurement of the wheel's spinning speed. To convert rpm to rad/s, the formula used is \[\begin{equation}\text{rad/s} = \text{rpm} \times \frac{2\pi}{60}\end{equation}\]With this conversion, it's easier to apply angular concepts to various problems, such as determining the angular acceleration of the wheel as it comes to a stop.

Radians Per Second

Radians per second (\text{rad/s}) provide a unit for angular velocity or angular speed, indicating the rotational distance traveled per second. Unlike degrees, radians are a natural choice for measuring angles in physics, especially when dealing with circular motion and rotational dynamics.

A radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. Since there are \[\begin{equation}2\pi\end{equation}\]radians in a full circle, a complete rotation corresponds to \[\begin{equation}2\pi \text{ rad}\end{equation}\]Therefore, when we say that a bicycle wheel rotates at 7.854 rad/s, it means that for every second, the wheel covers an angular distance of 7.854 radians. This precision is crucial when calculating other rotational quantities like angular acceleration.

Angular Kinematic Formula

The angular kinematic formula is a mathematical model that relates an object's angular velocity, angular acceleration, and time. It is analogous to the linear kinematic equations, which relate velocity, acceleration, and time for translational motion.

The basic form of the angular kinematic formula is given as:\[\begin{equation}\omega_{f} = \omega_{i} + \alpha \times t\end{equation}\]where \(\omega_{f}\) is the final angular velocity, \(\omega_{i}\) is the initial angular velocity, \(\alpha\) is the angular acceleration, and \(t\) is the time. For the example where a bicycle wheel decelerates to a stop:\[\begin{equation}\alpha = \frac{\omega_{f} - \omega_{i}}{t}\end{equation}\]This formula allows us to solve for the angular acceleration when we know the time taken for the object to go from its initial to its final angular velocity. It's a fundamental formula used in various physics problems involving rotational motion.