Circular motion
| English | Chinese | Pinyin |
|---|---|---|
| centripetal acceleration | 向心加速度 | xiàng xīn jiā sù dù |
| angular speed | 角速度 | jiǎo sù dù |
| linear speed | 线速度 | xiàn sù dù |
| radian | 弧度 | hú dù |
| tangent | 切线 | qiè xiàn |
| centripetal force | 向心力 | xiàng xīn lì |
| conical pendulum | 圆锥摆 | yuán zhuī bǎi |
| energy conservation | 能量守恒 | néng liàng shǒu héng |
| centrifugal | 离心 | lí xīn |
| inertia | 惯性 | guàn xìng |
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)?
v = rω = 2 × 3 = 6 m/s.
Linear speed and angular speed are related by v = r ______.
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)?
a = rω² = 2 × 9 = 18 m/s².
The centripetal acceleration of a particle in circular motion points:
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.
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:
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