Question

A flywheel completes \(42.3\) rev as it slows from an angular speed of \(1.44 \mathrm{rad} / \mathrm{s}\) to a complete stop. (a) Assuming constant acceleration, what is the time required for it to come to rest? (b) What is the angular acceleration? (c) How much time is required for it to complete the first one-half of the \(42.3\) rev?

Short answer

The time required for the flywheel to come to rest can be calculated using the above steps. The angular acceleration can then be determined by dividing the change in angular speed by the time, and the time for the first half of the revolutions can be found similarly by adapting the steps, replacing the total revolutions with the half revolutions in the formula

Step-by-step solution

Convert Total Revolutions to Radians

First, we need to convert the total given revolutions to radians since we're dealing with angular measures. One revolution equals \(2π\) radians. So, \(42.3\) revolutions equal \(42.3 * 2π\) radians.

Calculate Time to Stop

Next, we use the equation \(w = w0 + αt\) to calculate the time it takes for the flywheel to stop. Since the flywheel comes to a stop, \(w = 0\). Rearranging for \(t\) gives us \(t = (w - w0) / α\). But then, we don't know the value of \(α\). Instead, let's us use the displacement equation to compute \(t\), \(θ = w0 * t + 0.5 * α * t^2\).

Solve for Time

We need to solve this equation for \(t\). Since \(w = 0\) and the displacement \(θ = 42.3 * 2π\), the equation becomes \(42.3 * 2π = 1.44 * t + 0.5 * α * t^2\). It's a quadratic equation with the form \(at^2 + bt + c = 0\) where \(a = α/2, b = -w0, and c = -θ\). Using the quadratic formula \(t = [ -b ± sqrt(b^2 - 4* α* c) ] / 2a\), there are two possible values for \(t\) but we consider the positive one as time can't be negative,.

Calculate Angular Acceleration

After getting \(t\), we can find angular acceleration \(α\) by rearranging the equation from step 2 as \(α = (w - w0) / t\).

Calculate Time for Half Revolutions

We repeat Steps 2 and 3, but this time with the displacement \(θ = 21.15 * 2π\) (half of 42.3 rev). Solve the quadratic equation to find the time required for the first 21.15 rev.

Key concepts

Angular Acceleration

Angular acceleration is a measure of how quickly an object changes its angular velocity. It's defined as the rate of change of angular velocity with respect to time and is typically represented by the symbol \( \alpha \). Imagine you're on a merry-go-round that's starting to spin faster; the rate at which the merry-go-round picks up speed is its angular acceleration.

In our exercise, we calculate the angular acceleration by using the final angular velocity (which is zero since the flywheel comes to a stop), the initial angular velocity, and the time it took for this change to occur. Mathematically, it's expressed as \( \alpha = \frac{\Delta \omega}{\Delta t} \), where \( \Delta \omega \) is the change in angular velocity and \( \Delta t \) is the time interval for this change.

Angular Displacement

Angular displacement refers to the change in the angle as an object rotates about a point. It's the angle through which a point, line, or body has been rotated in a specified sense about a specified axis. When we talk about a flywheel completing 42.3 revolutions, we are discussing its angular displacement.

In the provided problem, converting revolutions to radians is essential because radians are the standard unit of angular displacement in physics. One complete revolution is equal to \( 2\pi \) radians, so the flywheel's angular displacement is \( 42.3 \times 2\pi \) radians. It's crucial for students to comprehend how to translate between different units of measurement to accurately describe rotational motion.

Quadratic Equation in Physics

Quadratic equations often appear in physics problems involving acceleration, motion, and forces. They have the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable.

In the flywheel scenario, we encounter a quadratic equation when we express angular displacement \( \theta \) in terms of time \( t \) with the equation \( \theta = \omega_0 \times t + 0.5 \times \alpha \times t^2 \). Here, \( \omega_0 \) is the initial angular speed, and \( \alpha \) is the angular acceleration. To find \( t \), we rearrange it to match the quadratic form and solve it using the quadratic formula, granting us the time required for the flywheel to complete its motion. As a final note, in physics problems, it's critical to dismiss the negative root (if it exists) because time cannot be negative.

rad/s to Revolutions Conversion

Radians per second (rad/s) and revolutions are units used to express angular velocity. However, to solve angular motion physics problems, we often need to convert between these units. One complete revolution is equivalent to \( 2\pi \) radians. Therefore, to convert rad/s to revolutions, you divide the angular velocity in rad/s by \( 2\pi \).

For example, if our flywheel has an angular velocity of \( 1.44 \) rad/s and we wish to express this in revolutions per second, we calculate \( 1.44 / 2\pi \) to get the value in revolutions. This conversion is pivotal in establishing a common ground for various calculations in angular motion problems, ensuring that all angular measures are consistent.