Physics: Uniform Circular Motion and Acceleration

Uniform Circular Motion

• Uniform circular motion is movement at a constant speed along a circular path, like a car in a traffic circle.
• Velocity vectors at any instant are tangent to the path (circle). All velocity vectors have the same magnitude but different directions.
• This changing direction means uniform circular motion is accelerated motion.

Instantaneous Acceleration

• To find instantaneous acceleration, examine velocity change over a short time interval (Δt).
• Choose a convenient origin (e.g., the center of the circle).
• Consider two positions (r₁ and r₂) separated by a short time interval (Δt). The difference between these position vectors is Δr = r₂ - r₁.
• Corresponding velocity vectors (v₁ and v₂) are also shown.
• The velocity vectors are always perpendicular to the position vectors.
• The angle between velocity vectors equals the angle between position vectors.
• The triangles formed by position and velocity vectors are similar.
• The ratio of magnitudes of the vectors Δv and Δr is equal to the ratio of v and r.

Acceleration Equation

• The magnitude of acceleration (a) is given by: a = v²/r.
• This formula shows that acceleration is directed towards the center of the circle, and is called centripetal.
• More specifically [magnitude of Δv] = v/r × [magnitude of Δr]
• The direction of the acceleration is perpendicular to the velocity.

Example: Centripetal Acceleration

• A real-world scenario is examined; astronauts in a gondola spinning at 15 m radius.
• If the gondola rotates 24 revolutions per minute, the centripetal acceleration is approximately 95 m/s².

Angular Motion in a Plane

• Angular displacement (θ) is measured in radians, degrees, or revolutions.
• One radian is defined by an arc length equal to the radius of a circle at the center.

Angular Speed (ω)

• Angular speed (ω) measures how quickly an object rotates. Units are in rad/s.
• ω = (θ₂ - θ₁)/ t
• where θ₁ and θ₂are angular displacements at different times

Angular Acceleration (α)

• Angular acceleration (α) is the rate of change of angular speed. Units are rad/s².
• α = (ω₂ - ω₁)/ t
• where ω₁ and ω₂are angular speeds at different times

Equations for Uniformly Accelerated Angular Motion

• Analogous to linear motion equations, but for angular motion.
• Formulas for average angular velocity, angular displacement, etc. are given.

Explore the concepts of uniform circular motion and instantaneous acceleration in this quiz. Understand how velocity vectors behave in circular paths and learn to calculate instantaneous acceleration over short time intervals. Test your knowledge of these fundamental principles of physics.