Understanding Angular Acceleration Through a Rotating Wheel: A Step-By-Step Analysis

Angular acceleration is a crucial concept in rotational motion, and this article provides a detailed exploration of how to calculate the constant angular acceleration of a wheel given specific conditions. We will walk through a series of steps, employing kinematic equations to solve for the angular acceleration, and provide a comprehensive solution.

Key Concepts: Angular Velocity, Angular Acceleration, and Rotational Motion

Before diving into the problem, it's important to grasp the fundamental concepts involved. Angular velocity ((omega)) is the rate of change of angular displacement, measured in radians per second (rad/s). Angular acceleration ((alpha)) is the rate of change of angular velocity, measured in radians per second squared (rad/s2). Rotational motion is the motion of an object around a fixed axis.

Problem Statement and Kinematic Equations

The problem we're addressing involves a wheel rotating through a specific number of revolutions over a given time, with a known final angular velocity. Our goal is to find the constant angular acceleration ((alpha)) of the wheel. Here are the given values:

Time ((t)): 3 seconds Total revolutions: 37 Final angular velocity ((omega_f)): 98 rad/s

Calculating the Total Angular Displacement ((theta))

The first step in solving the problem is to find the total angular displacement ((theta)) in radians. Since the wheel rotates through 37 revolutions, we can express the total angular displacement as follows:

(theta 37 times 2pi 74pi text{ radians})

Applying Kinematic Equations

There are two key kinematic equations we will use for rotational motion:

(omega_f omega_i alpha t) (theta omega_i t frac{1}{2} alpha t^2)

Step 1: Determine Initial Angular Velocity ((omega_i))

First, we rearrange the first equation to solve for (alpha):

(alpha frac{omega_f - omega_i}{t})

Step 2: Substitute into the Second Equation

Next, we substitute the expression for (alpha) into the second equation:

(theta omega_i t frac{1}{2} alpha t^2)

We express (theta) terms in terms of (omega_i) and (alpha):

(74pi omega_i t frac{1}{2} left(frac{omega_f - omega_i}{t}right) t^2)

This simplifies to:

(74pi omega_i t frac{1}{2} (t cdot (omega_f - omega_i)))

Simplifying further:

(74pi omega_i cdot 3 frac{1}{2} (98 - omega_i) times 3)

(74pi 3omega_i frac{3}{2} cdot 98 - frac{3}{2} omega_i)

(74pi 3omega_i 147 - frac{3}{2} omega_i)

(74pi frac{3omega_i - frac{3}{2} omega_i}{1} 147)

(74pi frac{3}{2} omega_i 147)

Step 3: Solve for (omega_i)

Now, we solve for (omega_i):

(frac{3}{2} omega_i 74pi - 147)

(omega_i frac{2}{3} (74pi - 147))

Using (pi approx 3.14), we find:

(74pi approx 232.2)

(omega_i frac{2}{3} (232.2 - 147))

(omega_i frac{2}{3} times 85.2 approx 56.8 text{ rad/s})

Step 4: Calculate Angular Acceleration ((alpha))

Finally, we find (alpha):

(alpha frac{omega_f - omega_i}{t} frac{98 - 56.8}{3})

(alpha frac{41.2}{3} approx 13.73 text{ rad/s}^2)

Conclusion

The constant angular acceleration of the wheel is approximately (13.73 text{ rad/s}^2).

Related Keywords

Angular acceleration Rotational motion Rotational kinematics

FAQs

Q: What is angular velocity?

A: Angular velocity ((omega)) is a measure of how fast an object rotates, often expressed in radians per second.

Q: What is angular acceleration?

A: Angular acceleration ((alpha)) measures the rate of change of angular velocity over time.

Q: How are angular displacement, angular velocity, and angular acceleration related?

A: Angular displacement ((theta)) is the total angle rotated, angular velocity ((omega)) is the rate of change of angular displacement, and angular acceleration ((alpha)) is the rate of change of angular velocity.