Centripetal Acceleration
Centripetal acceleration is the acceleration experienced by an object moving in a circular path, always directed toward the center of the circle. Although the object's speed may remain constant, its direction changes continuously, producing a non-zero acceleration perpendicular to the velocity. This acceleration is responsible for the continuous change in the direction of motion.
Key Formula
ac = v² / r = rω²
LaTeX: a_c = \frac{v^2}{r} = r\omega^2
| Symbol | Meaning | Unit |
|---|---|---|
| a_c | Centripetal acceleration | m/s² |
| v | Linear speed | m/s |
| r | Radius of circular path | m |
| \omega | Angular velocity | rad/s |
Worked Example
Problem
The Earth moves in a nearly circular orbit of radius 1.496 × 10¹¹ m around the Sun with a speed of 2.98 × 10⁴ m/s. Find the centripetal acceleration of the Earth.
Solution
Step 1: Identify given — v = 2.98 × 10⁴ m/s, r = 1.496 × 10¹¹ m. Step 2: Apply ac = v²/r. Step 3: ac = (2.98 × 10⁴)² / (1.496 × 10¹¹) = 8.88 × 10⁸ / 1.496 × 10¹¹.
Answer
ac ≈ 5.93 × 10⁻³ m/s², directed toward the Sun.
Centripetal acceleration for objects in circular motion
| Object | Radius (m) | Speed (m/s) | ac (m/s²) | Compared to g |
|---|---|---|---|---|
| Earth orbiting Sun | 1.50×10¹¹ | 2.98×10⁴ | 5.93×10⁻³ | 6.0×10⁻⁴ g |
| Moon orbiting Earth | 3.84×10⁸ | 1,022 | 2.72×10⁻³ | 2.8×10⁻⁴ g |
| Car on curve (r=50 m) | 50 | 20 | 8.0 | 0.82 g |
| Tip of fan blade | 0.30 | 15 | 750 | 76.5 g |
| F1 car cornering | 150 | 50 | 16.7 | 1.7 g |
Interactive Tools
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Related Terms
Centripetal Force
Centripetal force is the net inward force that acts on an object moving in a circular path, always directed toward the center of the circle. It is not a new type of force but rather the resultant of existing forces (tension, gravity, friction, normal force) that provides the necessary centripetal acceleration. Without this inward force, an object would continue in a straight line by Newton's first law.
Circular Motion
Circular motion is the motion of an object along the circumference of a circle or a circular path at a constant or varying speed. In uniform circular motion, the speed is constant but the velocity vector continuously changes direction, requiring a centripetal acceleration and force directed toward the center. Circular motion is fundamental to understanding planetary orbits, rotating machinery, and many natural phenomena.
Angular Velocity
Angular velocity is the rate of change of angular displacement of a rotating object with respect to time. It is a vector quantity whose direction is given by the right-hand rule along the axis of rotation. Angular velocity is the rotational analogue of linear velocity and is central to the analysis of rotating machinery, celestial bodies, and rigid body dynamics.
Derived from Latin "centrum" (center) and "petere" (to seek), combined with the Latin "accelerare" (to quicken). The concept was formally developed by Christiaan Huygens in 1659 in "De vi centrifuga".