Problem 23 - College Physics

Question

\(\cdot \mathrm{A}\) wheel rotates with a constant angular velocity of 6.00 \(\mathrm{rad} / \mathrm{s}\) (a) Compute the radial acceleration of a point 0.500 \(\mathrm{m}\) from the axis, using the relation \(a_{\mathrm{rad}}=\omega^{2} r .\) (b) Find the tangential speed of the point, and compute its radial acceleration from the relation \(a_{\text { rad }}=v^{2} / r\) .

Step-by-Step Solution

Answer

(a) Radial acceleration is 18.00 m/s². (b) Tangential speed is 3.00 m/s and radial acceleration is 18.00 m/s².

Step 1: Calculate Radial Acceleration Using Angular Velocity

First, we need to compute the radial acceleration using the formula \(a_{\mathrm{rad}}=\omega^{2} \cdot r\). We know that \(\omega = 6.00 \, \mathrm{rad/s}\) and \(r = 0.500 \, \mathrm{m}\). Substitute these values into the formula:\[a_{\mathrm{rad}} = (6.00)^2 \times 0.500 = 36.00 \times 0.500 = 18.00 \, \mathrm{m/s^2}\]Thus, the radial acceleration using angular velocity is \(18.00 \, \mathrm{m/s^2}\).

Step 2: Find Tangential Speed of the Point

The tangential speed, \( v \), of a point on a rotating wheel is given by the formula \( v = \omega \cdot r \). Using \(\omega = 6.00 \, \mathrm{rad/s}\) and \(r = 0.500 \, \mathrm{m}\), we have:\[v = 6.00 \times 0.500 = 3.00 \, \mathrm{m/s}\]Thus, the tangential speed of the point is \(3.00 \, \mathrm{m/s}\).

Step 3: Calculate Radial Acceleration Using Tangential Speed

Now, we use the tangential speed to compute the radial acceleration with the formula \(a_{\mathrm{rad}}=\frac{v^2}{r}\). Substitute \(v = 3.00 \, \mathrm{m/s}\) and \(r = 0.500 \, \mathrm{m}\) into the formula:\[a_{\mathrm{rad}} = \frac{(3.00)^2}{0.500} = \frac{9.00}{0.500} = 18.00 \, \mathrm{m/s^2}\]Thus, the radial acceleration calculated from the tangential speed is also \(18.00 \, \mathrm{m/s^2}\).

Key Concepts

Angular Velocity

Angular velocity is crucial in understanding how objects rotate. It tells us how quickly the angle changes as an object spins. Measured in radians per second (\( ext{rad/s}\)), angular velocity essentially transforms linear thinking into a rotating perspective.
- **Definition:** It's the rate of change of angular displacement and is a vector quantity, meaning it has both magnitude and direction.- **Constant Angular Velocity:** When this velocity doesn't change over time, the motion is uniform.- **Importance in Calculations:** In the exercise, angular velocity is used to derive radial acceleration since \(a_{\text{rad}} = \omega^2 \cdot r\). If the wheel spins faster, radial acceleration increases as well.
Understanding angular velocity allows predictions of how quickly a rotating system behaves and is foundational for many realms of physics, from mechanical to celestial dynamics.

Tangential Speed

Tangential speed helps describe the motion of a point on the edge of a rotating object. Unlike angular velocity, which remains constant for all points on a rotating body, tangential speed varies based on the distance from the axis of rotation. - **Definition:** It can be thought of as the linear speed of a point revolving along a circular path.- **Relation to Angular Velocity:** As derived in the original exercise, tangential speed is calculated by \(v = \omega \cdot r\). Here, \(\omega\) is the angular velocity, showing a direct multiplication relationship with the radius.- **Practical Implications:** If you're in a car moving around a corner, your tangential speed affects how "fast" you feel the turn. For understanding rotating systems, visualizing tangential speed alongside angular velocity gives a complete picture of motion.

Rotational Motion

Rotational motion describes the movement of objects around a center or axis. It is central to understanding how various systems, from wheels to solar systems, behave and function.
- **Axis of Rotation:** The imaginary line through the center of this motion.- **Components of Rotational Motion:** They include angular velocity, angular acceleration, and rotational kinetic energy.- **Radial Acceleration:** This symbolizes the movement directed towards the center in circular paths, crucial for keeping objects in rotation without spiraling off. Two important equations describe it: - Using angular velocity: \(a_{\text{rad}} = \omega^2 \cdot r\). - Using tangential speed: \(a_{\text{rad}} = \frac{v^2}{r}\).The analysis of rotational motion involves understanding how these components interact to affect the movement of objects. Mastery in these areas enables predicting, designing, and operating systems involving rotational dynamics.