Circular motion
Introduced| English | Meaning |
|---|---|
| centripetal acceleration/senˈtrɪpɪtl əkˌseləˈreɪʃn/ | The acceleration towards the centre of a circular path, caused by the changing direction of velocity. |
| angular speed/ˈæŋɡjʊlə spiːd/ | The rate at which angular position changes, measured as angle per unit time. |
| linear speed/ˈlɪnɪə spiːd/ | The rate at which distance is travelled along a path. |
| radian/ˈreɪdɪən/ | The angle subtended at a circle's centre by an arc equal in length to its radius. |
| tangent/ˈtændʒənt/ | A straight line that touches a curve at a point and has the same direction there. |
| centripetal force/senˈtrɪpɪtl fɔːs/ | The resultant force towards the centre of a circular path. |
| conical pendulum/ˈkɒnɪkl ˈpendjʊləm/ | A pendulum whose bob moves in a horizontal circle while its string sweeps out a cone. |
| energy conservation/ˈenədʒi ˌkɒnsəˈveɪʃn/ | energy conservation |
| centrifugal/ˌsentrɪˈfjuːɡl/ | centrifugal |
| inertia/ɪˈnɜːʃə/ | An object's resistance to changes in its state of motion, measured by its mass in translational motion. |
Why you feel pushed out on a roundabout
- Whirl a ball on a string and it pulls outward — yet the force on the ball points inward.
- Anything moving in a circle is constantly accelerating toward the centre, even at steady speed, because its direction keeps changing.
Angular speed 角速度 and linear speed 线速度
- Angular speed $\omega$ (radians 弧度 per second) links to linear speed by:
- A point further from the centre (bigger $r$) moves faster for the same $\omega$.
In circular motion velocity is tangent 切线 and acceleration points to the centre
Angle in radians
Circular motion is measured in radians: drag the angle θ and radius r to see the arc swept — angular speed ω turns this into v = rω.
A particle moves in a circle of radius 2 m with angular speed 3 rad/s. What is its speed v = rω (m/s)?
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v = rω = 2 × 3 = 6 m/s.
Linear speed and angular speed are related by v = r ______.
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v = rω.
Centripetal acceleration 向心加速度
- The acceleration points to the centre — centripetal acceleration:
- It is caused by a centripetal force 向心力 $F = ma$ (tension, friction, gravity, or a normal reaction).
Velocity is tangent to the circle; the acceleration (and net force) point to the centre.
For the same particle (r = 2, ω = 3), what is the centripetal acceleration a = rω² (m/s²)?
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a = rω² = 2 × 9 = 18 m/s².
The centripetal acceleration of a particle in circular motion points:
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It points to the centre — that is what continually changes the direction of motion.
The velocity of a particle in circular motion is directed along the tangent to the circle.
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Velocity is always tangent; the acceleration is perpendicular to it, toward the centre.
Worked example
- A particle moves in a circle of radius $2$ m at angular speed $3\ \tfrac{\text{rad}}{\text{s}}$.
- Speed $v = r\omega = 2 \times 3 = 6\ \tfrac{\text{m}}{\text{s}}$.
- Centripetal acceleration $a = r\omega^2 = 2 \times 3^2 = 18\ \tfrac{\text{m}}{\text{s}^2}$.
Horizontal vs vertical circles
- Horizontal circle (e.g. a conical pendulum 圆锥摆): speed is constant.
- Vertical circle: speed changes with height, so combine circular motion with energy conservation 能量守恒.
There is no outward "centrifugal 离心 force" on the particle. The real force is the inward (centripetal) one; the outward feeling is your body's inertia 惯性 resisting the change in direction.
- In circular motion the normal contact force or tension provides the centripetal force.
For motion in a vertical circle, the speed is not constant, so you also use:
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Height changes, so gravitational PE ↔ KE: use energy conservation alongside the circular-motion equations.
You've got it
- $v = r\omega$; centripetal acceleration $a = r\omega^2 = \dfrac{v^2}{r}$, directed to the centre
- a real inward centripetal force causes it ($F = ma$)
- horizontal circle → constant speed; vertical circle → use energy conservation