Newton's Second Law
Introduced| English |
|---|
| mass/mæs/ |
| Newton's second law/ˈnjuːtnz ˈsekənd lɔː/ |
| net force/net fɔːs/ |
| acceleration/əkˌseləˈreɪʃn/ |
| inertia/ɪˈnɜːʃə/ |
| components/kəmˈpəʊnənts/ |
| pulley/ˈpʊli/ |
The same shove -- a feather flies, a fridge barely budges
- Give a feather a gentle push and it shoots off; push a fridge just as hard and it hardly moves.
- Same force, wildly different results -- the difference is mass 质量.
- One equation ties force, mass, and motion together.
- It is the engine of mechanics: give it the forces and the mass, and it hands you the motion.
Newton's second law
- Newton's second law 牛顿第二定律 states:
- The net force 合力 on an object equals its mass times its acceleration 加速度.
- Rearranged, $\vec{a} = \dfrac{\vec{F}_{net}}{m}$ -- the forces set the acceleration.
A net force of $20\ \text{N}$ acts on a $4\ \text{kg}$ cart. What is its acceleration (in $\tfrac{\text{m}}{\text{s}^2}$)?
$a = \dfrac{F_{net}}{m} = \dfrac{20}{4} = 5\ \tfrac{\text{m}}{\text{s}^2}$.
Rearranging the second law: acceleration equals the net force divided by the ____.
$\vec{a} = \dfrac{\vec{F}_{net}}{m}$ -- the net force over the mass.
More force, less mass
- The one equation carries two rules:
- More net force → proportionally more acceleration.
- More mass → proportionally less acceleration (mass resists change -- its inertia 惯性).
- That is exactly why the light feather leaps and the massive fridge barely stirs.
Keep the force the same but double the mass. The acceleration...
Since $a = F_{net}/m$, doubling $m$ halves $a$ -- mass and acceleration are inversely proportional.
Select all changes that would increase an object's acceleration.
More net force or less mass both raise $a = F_{net}/m$; more mass lowers it, and cancelling forces gives zero net force.
It is the net force that counts
- An object usually feels several forces at once -- a push, gravity, friction.
- Only their vector sum, the net force, drives the acceleration.
- If the forces cancel to zero, there is no acceleration: the object stays at rest or keeps constant velocity.
If all the forces on an object cancel, it cannot be moving.
Zero net force means zero acceleration, not zero velocity. A moving object simply keeps its constant velocity.
Splitting forces into components
- When forces point in awkward directions (say, on a ramp), split each into components 分量 along chosen axes.
- Apply $\sum F = ma$ separately in each direction.
- For connected objects on a pulley 滑轮, write the law for each mass and solve together.
Free-body diagram: F = ma
The resultant of the forces, divided by the mass, gives the acceleration. Change a force and watch the acceleration respond.
A $2\ \text{kg}$ box is pulled with $30\ \text{N}$ while friction drags back $10\ \text{N}$. Find its acceleration (in $\tfrac{\text{m}}{\text{s}^2}$).
Net force $= 30 - 10 = 20\ \text{N}$, so $a = 20/2 = 10\ \tfrac{\text{m}}{\text{s}^2}$. Use the net force, not the $30\ \text{N}$ alone.
A net force of $20\ \text{N}$ acts on a $5\ \text{kg}$ box.
- Acceleration: $a = \dfrac{F_{net}}{m} = \dfrac{20}{5} = 4\ \tfrac{\text{m}}{\text{s}^2}$.
- Double the mass to $10\ \text{kg}$ with the same force, and the acceleration halves to $2\ \tfrac{\text{m}}{\text{s}^2}$.
Use the net force, not one force you happen to notice. A box pulled with $30\ \text{N}$ while friction drags back $10\ \text{N}$ has a net force of $30 - 10 = 20\ \text{N}$ -- so $a = 20/m$, not $30/m$.
Newton's second law is $\vec{F}_{net} = m\vec{a}$, or $\vec{a} = \dfrac{\vec{F}_{net}}{m}$: acceleration equals the net force divided by the mass. More force means more acceleration; more mass means less. Always add the forces as vectors first, then divide.