Uniform circular motion has an acceleration even though the speed is constant, because the direction of the velocity changes at every instant.
That acceleration always points toward the center of the circle; it is called centripetal acceleration, and its size is a = v²/r = rω².
If the centripetal force pulling the object toward the center disappears, the object flies off in a straight line along the tangent at that instant.
In the picture, changing the radius and the angular speed changes the lengths of the tangential velocity arrow and the center-pointing acceleration arrow according to v = rω and a = rω².
Circular Motion Is Accelerated Motion
Even when the speedometer reading stays the same, the velocity of an object going around a circle does not. Velocity is a vector with both size and direction, and on a circle the tangent points a different way at every spot, so the velocity changes from moment to moment. Leave that change out of the account and you cannot explain a satellite circling the Earth or a ball on a string tracing a circle. Knowing the radius r and the angular speed ω, the linear speed is v = rω, the centripetal acceleration is a = v²/r = rω², and the period for one turn is T = 2π/ω. The center-pointing net force that produces this acceleration is the centripetal force F = ma, and these three relations are the starting point for treating a change of direction as accelerated motion.
🎡 Why Is There Acceleration When the Speed Is Constant?
①In uniform circular motion the speed stays the same, but the direction of the velocity keeps changing along the circle
②Velocity includes direction, so a change in direction is a change in velocity, and that means there is an acceleration
③This acceleration always points toward the center of the circle, so it is called centripetal acceleration
④If the centripetal force is cut off, the object moves straight on in the direction of its velocity at that instant, along the tangent
⑤Gravity holding a satellite, the tension in a string swinging a ball and the friction on the tires of a car rounding a curve all act as centripetal forces
Drawing Velocity and Centripetal Acceleration
4 m
1.5 rad/s
seegongsik.com
One instant of an object (green dot) moving counterclockwise around a circular path. The orange arrow is the velocity v along the tangent, the red arrow is the centripetal acceleration a_c toward the center O, and each length is proportional to its size
The circle in the middle is the path, and the green dot at the lower right is the position, at one instant, of an object moving counterclockwise. The orange arrow reaching up and to the right from the dot is the velocity v along the tangent, the red arrow reaching toward the center O is the centripetal acceleration a_c, and the dashed line is the radius r. Each arrow is drawn at a fixed scale for its own quantity, so you can compare how much each grows as the sliders move. Moving the radius slider between 1 m and 6 m enlarges the circle and both arrows together; moving the angular speed slider between 0.5 rad/s and 3 rad/s leaves the circle the same size and only lengthens the arrows, with the red arrow growing much faster than the orange one. That is because the angular speed enters a = rω² squared.
💡 Reading the Picture
①The velocity v always points along the tangent to the path, so it is perpendicular to the radius
②The centripetal acceleration a_c always points along the radius toward the center O
③Increasing the radius or the angular speed makes v = rω and a = rω² grow together, and doubling the angular speed doubles v and quadruples a_c
Formulas for Uniform Circular Motion
A common belief is that constant speed means zero acceleration. But acceleration is how fast the velocity changes with time, and velocity includes direction, so the acceleration is not zero while the object goes around the circle. Another frequent mistake is drawing the centripetal force along the direction of travel; the centripetal force always points to the center, perpendicular to the motion, so it cannot change the speed and only turns the direction. When a string snaps, the object is not flung outward; it moves straight on along the tangent at that instant. So before putting numbers into a formula, first sort out which arrow is tangential and which one points to the center.
Centripetal Acceleration
a = v²r = rω²
The size of the acceleration toward the center: the speed squared divided by the radius, which equals the radius times the angular speed squared
Linear Velocity
v = rω
Linear speed (m/s) = radius (m) × angular speed (rad/s)
Period and Frequency
T = 2πω = 2πrv, f = 1T = ω2π
The period T is the time for one turn, and the frequency f is the number of turns in 1 s, with f = 1/T
Centripetal Force
F = ma = mv²r = mrω²
The net force toward the center needed to keep the circular motion going
Worked Examples
Example 1
An object in uniform circular motion on a circle of radius 2 m moves at a speed of 4 m/s. What is its centripetal acceleration?
1
Use the centripetal acceleration formula a = v²/r.
a = v2r
2
Substitute v = 4 m/s and r = 2 m.
a = 422 = 8 m/s2
▸ 8 m/s² (toward the center)
At the same 2 m radius, doubling the speed to 8 m/s makes the centripetal acceleration four times as large, 32 m/s².
Example 2
A 0.5 kg object tied to a 1 m string moves in a horizontal circle with angular speed 2 rad/s. What is the tension in the string?
1
The tension in the string is the centripetal force, so use F = mrω².
F = mrω2
2
Substitute m = 0.5 kg, r = 1 m and ω = 2 rad/s.
F = 0.5 × 1 × 22 = 2 N
▸ 2 N
Use F = mrω² when the angular speed is given and F = mv²/r when the linear speed is given. Here the linear speed is v = 1 × 2 = 2 m/s, so 0.5 × 2²/1 = 2 N gives the same answer.
Summary
Centripetal acceleration
a = v²r
Linear velocity
v = rω
CSAT-style problem, adapted
In uniform circular motion, if the radius is kept fixed and the angular speed is doubled, by what factor does the centripetal acceleration change?
①2×
②4×
③12×
④No change
⑤8×
▸ ② 4×
1
The centripetal acceleration is a = rω², so with the same radius it is proportional to the square of the angular speed.
a = rω2
2
If ω becomes 2 times larger, ω² becomes 4 times larger, so a also becomes 4 times larger.
ω→2ω ⇒ a ∝ (2ω)2 = 4rω2
🎯 Exam Points
①Uniform circular motion is accelerated motion: the speed is constant but the direction of the velocity changes
②The centripetal acceleration a = v²/r = rω² always points to the center
③If the centripetal force F = mv²/r disappears, the object moves straight on along the tangent
④v = rω, T = 2π/ω, f = 1/T = ω/(2π)
⑤Examples of centripetal force: gravity (satellite), tension (ball on a string), friction (car on a curve)