Solved examples of finding linear acceleration using angular acceleration

📚 Quick Study Guide

• 🔄 Angular Acceleration ($\alpha$): The rate of change of angular velocity, measured in rad/s².
• 📏 Linear Acceleration ($a$): The rate of change of linear velocity, measured in m/s².
• 🔗 Relationship: Linear acceleration ($a$) is related to angular acceleration ($\alpha$) by the radius ($r$) of the circular path: $a = r\alpha$.
• 📐 Tangential Acceleration: This is the component of linear acceleration tangent to the circular path. When using $a = r\alpha$, we're finding the tangential acceleration.
• 💡 Units: Always ensure consistent units! Radius in meters (m), angular acceleration in rad/s², and linear acceleration in m/s².
• 📝 Direction: The direction of linear acceleration is tangent to the circle, while angular acceleration is along the axis of rotation.

Practice Quiz

1. A wheel with a radius of 0.5 m has a constant angular acceleration of 2 rad/s². What is the linear acceleration of a point on the edge of the wheel?

   1. 0.5 m/s²
   2. 1 m/s²
   3. 1.5 m/s²
   4. 2 m/s²

2. A rotating disk has an angular acceleration of 3 rad/s² and a radius of 0.25 m. Find the linear acceleration at the edge of the disk.

   1. 0.25 m/s²
   2. 0.50 m/s²
   3. 0.75 m/s²
   4. 1.00 m/s²

3. If a point on a rotating object 0.3 m from the center has a linear acceleration of 1.2 m/s², what is the angular acceleration of the object?

   1. 2 rad/s²
   2. 3 rad/s²
   3. 4 rad/s²
   4. 5 rad/s²

4. A pulley with a radius of 0.1 m accelerates from rest to an angular velocity of 5 rad/s in 2 seconds. What is the linear acceleration of a point on the rim of the pulley during this time?

   1. 0.1 m/s²
   2. 0.25 m/s²
   3. 0.5 m/s²
   4. 1 m/s²

5. A spinning top has an angular acceleration of 8 rad/s². If a point 0.05 m from the center is considered, what is its linear acceleration?

   1. 0.2 m/s²
   2. 0.4 m/s²
   3. 0.6 m/s²
   4. 0.8 m/s²

6. A CD with a radius of 0.06 m speeds up with an angular acceleration of 5 rad/s². Calculate the linear acceleration of a point on its outer edge.

   1. 0.1 m/s²
   2. 0.3 m/s²
   3. 0.5 m/s²
   4. 0.7 m/s²

7. A motor's shaft with a radius of 0.04 m has a constant angular acceleration of 10 rad/s². Find the linear acceleration on the surface of the shaft.

   1. 0.2 m/s²
   2. 0.4 m/s²
   3. 0.6 m/s²
   4. 0.8 m/s²