An object moving at constant speed along a circle — but at every instant, the direction of velocity changes. So it has acceleration! This is "centripetal acceleration" — always toward the center.
Definition
- Constant speed:
|\vec v| = v = const - Path: circle of radius
r - Period
T: time for one full revolution - Speed:
v = 2\pi r / T
Why is there acceleration?
Velocity is a vector — magnitude and direction. If direction changes (even with constant magnitude), the velocity vector is changing — so its derivative (acceleration) is nonzero.
Magnitude of centripetal acceleration
\[ \boxed{a_c = \frac{v^2}{r} = \frac{4\pi^2 r}{T^2}} \]
Direction: always toward the center of the circle — perpendicular to velocity (which is tangent to the circle).
Derivation
Velocity at time t:
\[ \vec v(t) = -v \sin(\omega t)\hat i + v \cos(\omega t)\hat j \]
with \omega = 2\pi/T the angular velocity.
Differentiate for acceleration: \[ \vec a(t) = -v\omega \cos(\omega t)\hat i - v\omega \sin(\omega t)\hat j \]
Magnitude: |\vec a| = v\omega = v^2/r ✓
Direction: opposite the position vector (which points outward). So it points inward.
Examples
Example 1 — car in a curve:
Car at 20 m/s on a curve of radius 100 m. Centripetal acceleration:
\[ a_c = \frac{20^2}{100} = 4\ m/s^2 \]
Example 2 — Moon around Earth:
Orbital radius r = 3.84 \times 10^8 m, period T = 27.3\ \text{days} = 2.36 \times 10^6\ s.
\[ a_c = \frac{4\pi^2 \cdot 3.84 \times 10^8}{(2.36 \times 10^6)^2} \approx 2.7 \times 10^{-3}\ m/s^2 \]
Matches Earth's gravitational acceleration at Moon's distance (Gravity Article 1).
Example 3 — astronaut centrifuge:
A seat at radius 5 m spinning once per second (T = 1 s).
\[ a_c = \frac{4\pi^2 \cdot 5}{1^2} \approx 197\ m/s^2 \approx 20g \]
Extreme! Astronauts endure ~9g in training simulators.
Period, frequency, angular velocity
- Period
T: seconds per revolution - Frequency
f = 1/T: revolutions per second (hertz, Hz) - Angular velocity
\omega = 2\pi/T = 2\pi f: radians per second
Relations:
\[ v = \omega r,\quad a_c = \omega^2 r = v^2/r \]
A few notes and common mistakes
1. Constant speed ≠ zero acceleration. Most common mistake. If direction changes, there is acceleration.
2. Centripetal points inward. "Centripetal" means "center-seeking". Not outward (that would be "centrifugal" — itself a common misconception).
3. Velocity is tangent to the circle. Perpendicular to the radius. If released, an object flies off tangentially (like a ball off a rope).
4. Angular units.
In physics, always use radians (not degrees). 2\pi radians = one full turn.
What you should be able to do
- Compute
a_c = v^2/rand know direction is inward - Relate
v, \omega, T, f - Express
a_cinm/s^2org - Recognize why constant speed doesn't imply zero acceleration
Preview of §4.5
Interesting question: speed relative to whom? A ball falls in a train — vertical to the passenger, parabolic to a ground observer. §4.5 covers "relative velocity".
📚 See also: Halliday Vol 1, Ch 4, §4.5 — Uniform Circular Motion.
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