Problem 23 - University Physics with Modern Physics

Question

A wheel of diameter 40.0 \(\mathrm{cm}\) starts from rest and rotates with a constant angular acceleration of 3.00 \(\mathrm{rad} / \mathrm{s}^{2}\) . At the instant the wheel has computed its second revolution, compute the radial acceleration of a point on the rim in two ways: (a) using the relationship \(a_{\mathrm{rad}}=\omega^{2} r\) and \((\mathrm{b})\) from the relationship \(a_{\mathrm{red}}=v^{2} / r\)

Step-by-Step Solution

Answer

The radial acceleration is 15.06 m/s² for both methods.

Step 1: Find the Angular Displacement for Two Revolutions

Since one complete revolution equals \(2\pi\) radians, two revolutions correspond to \(4\pi\) radians. Therefore, the angular displacement \(\theta = 4\pi\).

Step 2: Use Angular Displacement to Find Final Angular Velocity

Use the kinematic equation \(\omega_f^2 = \omega_i^2 + 2\alpha\theta\), where \(\omega_i=0\) (since the wheel starts from rest), \(\alpha=3.00\, \mathrm{rad/s^2}\), and \(\theta = 4\pi\). Solving for \(\omega_f\) gives \(\omega_f = \sqrt{2 \cdot 3.00 \cdot 4\pi} = \sqrt{24\pi} \approx 8.68\, \mathrm{rad/s}\).

Step 3: Calculate Radial Acceleration Using Angular Velocity

The radial acceleration \(a_{\mathrm{rad}}\) in terms of angular velocity is given by \(a_{\mathrm{rad}} = \omega^2 r\). The radius \(r = \frac{40.0}{2} = 20.0\, \mathrm{cm} = 0.2\, \mathrm{m}\). So, \(a_{\mathrm{rad}} = (8.68^2) \cdot 0.2 = 15.06\, \mathrm{m/s^2}\).

Step 4: Calculate Tangential Velocity

Tangential velocity \(v\) is related to angular velocity by \(v = \omega r\). Thus, \(v = 8.68 \cdot 0.2 = 1.736\, \mathrm{m/s}\).

Step 5: Calculate Radial Acceleration Using Tangential Velocity

The radial acceleration \(a_{\mathrm{rad}}\) in terms of tangential velocity is given by \(a_{\mathrm{rad}} = \frac{v^2}{r}\). Substituting the values, \(a_{\mathrm{rad}} = \frac{1.736^2}{0.2} = 15.06\, \mathrm{m/s^2}\).

Key Concepts

Radial Acceleration

Radial acceleration, also known as centripetal acceleration, occurs when an object moves in a circular path. It is always directed towards the center of the circle. This force keeps the object moving in a curve instead of traveling in a straight line. Radial acceleration can be calculated using either the angular velocity or the tangential velocity.

For angular velocity, the formula is:

• \[ a_{\mathrm{rad}} = \omega^2 r \]

where \( \omega \) is the angular velocity and \( r \) is the radius of the circle.

For tangential velocity, it is:

• \[ a_{\mathrm{rad}} = \frac{v^2}{r} \]

This illustrates how radial acceleration can derive from different aspects of rotational motion.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or spins around a central point. It is typically expressed in radians per second (rad/s). For a wheel or any circular object, angular velocity indicates how many radians of angle it traverses per second as it rotates.

When an object starts from rest, like a wheel initially, its angular velocity starts at zero and increases based on the angular acceleration. The final angular velocity \( \omega_f \) can be determined from angular displacement \( \theta \) and angular acceleration \( \alpha \):

• \[ \omega_f^2 = \omega_i^2 + 2\alpha\theta \]

This equation showcases the relationship between initial angular velocity, angular acceleration, and angular displacement.

Tangential Velocity

Tangential velocity refers to the linear speed of any point on a rotating body, such as the rim of a spinning wheel. It is a representation of how fast a point moves along the circular path. Unlike angular velocity, which is rotation centered, tangential velocity is concerned with motion along the path of the circle.

The formula linking tangential velocity \( v \) to angular velocity \( \omega \) is:

• \[ v = \omega r \]

Here, \( r \) represents the radius of the rotation. As angular velocity changes, so does tangential velocity, illustrating how they are interconnected.

Kinematic Equations in Rotational Motion

Kinematic equations describe the motion of objects without considering the causes of this motion. In the case of rotational motion, these equations are analogous to those used in linear motion but adapted for rotation. They help calculate variables like angular displacement, angular velocity, and angular acceleration.

These equations include:

• Angular displacement: \[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
• Final angular velocity: \[ \omega_f = \omega_i + \alpha t \]
• Final angular velocity (squared): \[ \omega_f^2 = \omega_i^2 + 2\alpha\theta \]

These formulas provide a framework for solving rotational motion problems by converting linear motion concepts to a rotational context.