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define-angular-velocity-and-linear-velocity

๐Ÿš€ Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. In uniform circular motion, an object moves in a circular path at a constant speed. Two important concepts in this context are angular velocity and linear velocity. Angular velocity refers to the rate of change of the angle with respect to time, while linear velocity is the rate of change of the object's position along the circular path. Both velocities are related but represent different aspects of motion.

Theory Explanation

Understanding Angular Velocity

Angular velocity (ฯ‰) is defined as the angle (ฮธ) rotated per unit time (t). It is measured in radians per second (rad/s). The formula for angular velocity is ฯ‰ = ฮธ/t, where ฮธ is in radians and t is in seconds.

\[ \omega = \frac{\theta}{t} \]
Understanding Linear Velocity

Linear velocity (v) is the rate at which an object moves along the circular path. It is calculated using the formula v = rฯ‰, where r is the radius of the circular path and ฯ‰ is the angular velocity. Linear velocity is measured in meters per second (m/s).

\[ v = r\omega \]
Relationship Between Angular and Linear Velocity

The relationship between angular velocity and linear velocity can be expressed as v = rฯ‰. This means that the linear velocity is directly proportional to both the radius of the circular path and the angular velocity. As the radius increases, the linear velocity increases, provided the angular velocity remains constant.

\[ v = r\omega \]

Key Points

  • ๐ŸŽฏ Angular velocity is measured in radians per second (rad/s).
  • ๐ŸŽฏ Linear velocity is measured in meters per second (m/s).
  • ๐ŸŽฏ The relationship between linear and angular velocity is given by v = rฯ‰.

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Examples:💡

A car is moving in a circular path with a radius of 10 meters and completes one full revolution in 20 seconds. Calculate the angular velocity and linear velocity of the car.

Solution:

Step 1: First, calculate the angular velocity (ฯ‰). Since one full revolution is 2ฯ€ radians, we have ฮธ = 2ฯ€ and t = 20 seconds. Thus, ฯ‰ = ฮธ/t = 2ฯ€/20 = ฯ€/10 rad/s.

\[ \omega = \frac{2\pi}{20} = \frac{\pi}{10} \text{ rad/s} \]

Step 2: Next, calculate the linear velocity (v) using the formula v = rฯ‰. Here, r = 10 meters and ฯ‰ = ฯ€/10 rad/s. Therefore, v = 10 * (ฯ€/10) = ฯ€ m/s.

\[ v = 10 \cdot \frac{\pi}{10} = \pi \text{ m/s} \]

A wheel with a radius of 0.5 meters rotates with an angular velocity of 4 rad/s. Find the linear velocity of a point on the edge of the wheel.

Solution:

Step 1: Use the formula v = rฯ‰. Here, r = 0.5 meters and ฯ‰ = 4 rad/s. Thus, v = 0.5 * 4 = 2 m/s.

\[ v = 0.5 \cdot 4 = 2 \text{ m/s} \]

Common Mistakes

  • Mistake: Confusing angular velocity with linear velocity. Students often think they are the same, but they represent different aspects of motion.

    Correction: Remember that angular velocity is about the rate of rotation (in radians per second), while linear velocity is about the speed along the circular path (in meters per second).

  • Mistake: Incorrectly applying the formula v = rฯ‰. Some students forget to convert units or miscalculate the radius.

    Correction: Always check that the radius is in the correct units (meters) and that angular velocity is in radians per second before using the formula.

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