Circular Motion
| English | Chinese | Pinyin |
|---|---|---|
| centripetal | 向心的 | xiàng xīn de |
Whirl a ball on a string — where does it go if it snaps?
- Spin a ball on a string in a circle at steady speed. Now imagine the string snaps.
- It does not fly outward — it shoots off along the tangent, in a straight line.
- So something must constantly pull it inward to bend its path into a circle.
- That inward pull is the key to all circular motion.
Constant speed, changing velocity
- In uniform circular motion the speed is constant but the velocity is not.
- Velocity is a vector, and its direction changes every instant around the circle.
- A changing velocity means there is an acceleration — even at constant speed.
- This acceleration points toward the centre, so we call it centripetal 向心的.
A ball whirls on a string in a horizontal circle. The string suddenly snaps. Which way does the ball fly?
With no inward force, inertia carries the ball in a straight line — along the tangent where the string broke.
An object in uniform circular motion has zero acceleration because its speed is constant.
Its direction changes, so its velocity changes — there is a centripetal acceleration $v^2/r$ toward the centre.
Centripetal acceleration and force
- The centripetal acceleration has size $a_c = \dfrac{v^2}{r}$, directed to the centre.
- By Newton's second law, a net centripetal force $F_c = \dfrac{mv^2}{r}$ must point inward.
- Faster spin or a tighter circle both demand a larger inward force.
- This force is not new — it is provided by tension, gravity, friction, or a normal force.
Around the circle
Sweep the angle and change the radius to see how position and direction change around a circle.
A $2\ \text{kg}$ ball moves at $4\ \tfrac{\text{m}}{\text{s}}$ in a circle of radius $8\ \text{m}$. What centripetal force is needed, in $\text{N}$?
$F_c = \dfrac{mv^2}{r} = \dfrac{2 \times 16}{8} = 4\ \text{N}$, toward the centre.
In which direction does the net (centripetal) force point?
The net force is centripetal — it points toward the centre, bending the path into a circle.
Centripetal acceleration has magnitude $v^2 / \_\_$ (fill in the missing symbol).
$a_c = v^2/r$: bigger speed or smaller radius means a larger centripetal acceleration.
Select all forces that can provide the centripetal force in real situations.
Tension, gravity and friction can each supply the inward force. There is no real outward centrifugal force.
There is no outward "centrifugal" force
- The feeling of being flung outward in a turning car is your inertia, not a real force.
- Your body "wants" to go straight (tangent); the car door pushes you inward to turn you.
- The only real force on you is that inward push — there is no outward force acting.
- Call the inward net force centripetal; the outward "centrifugal force" is a fiction.
If the string breaks, the ball flies off along the tangent, not radially outward. And there is no real outward "centrifugal force" — the outward feeling is just inertia resisting the inward turn. The genuine net force is inward.
A $2\ \text{kg}$ ball moves at $4\ \tfrac{\text{m}}{\text{s}}$ in a circle of radius $8\ \text{m}$.
- $F_c = \dfrac{mv^2}{r} = \dfrac{2 \times 4^2}{8} = \dfrac{32}{8} = 4\ \text{N}$, directed toward the centre.
The string tension must supply this $4\ \text{N}$ inward.
Uniform circular motion has constant speed but changing velocity, so there is a centripetal acceleration $a_c = v^2/r$ toward the centre. A net centripetal force $F_c = mv^2/r$ (from tension, gravity, friction...) points inward. There is no real outward "centrifugal" force — that is inertia.