Круговое движение

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

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.

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.

• In circular motion the normal contact force or tension provides the centripetal force.

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