Spring Forces
Introduced| English | Meaning |
|---|---|
| Hooke's law/hʊks lɔː/ | The rule that extension is proportional to applied force while a spring remains within its limit of proportionality. |
| restoring force/rɪˈstɔːrɪŋ fɔːs/ | A force acting towards an object's equilibrium position after displacement. |
| extension/ekˈstenʃn/ | The increase in length of an object. |
| spring constant/sprɪŋ ˈkɒnstənt/ | The force required per unit extension of a spring in its linear range. |
| elastic limit/ɪˈlæstɪk ˈlɪmɪt/ | The greatest stress or deformation a material can experience and still return to its original shape when unloaded. |
Pull twice as far, feel twice the pull
- Stretch a spring a little and it tugs back gently; stretch it twice as far and it tugs twice as hard.
- The pull grows in exact step with how far you stretch — a beautifully simple rule.
- That rule is Hooke's law 胡克定律, and it describes springs, rubber, and many materials.
- The spring always pulls back toward its natural length — a restoring force 回复力.
Hooke's law
- The spring force is proportional to the extension 伸长量 $x$ from its natural length.
- In size: $F = kx$. The constant $k$ is the spring constant 弹簧劲度系数.
- $k$ measures stiffness — a stiff spring has a large $k$ (needs more force per metre).
- Its units are $\tfrac{\text{N}}{\text{m}}$: newtons of force for each metre of stretch.
Stretch the spring
Change the force and watch the extension grow in proportion — the slope is the spring constant.
A spring with $k = 200\ \tfrac{\text{N}}{\text{m}}$ is stretched by $0.10\ \text{m}$. What is the spring force, in $\text{N}$?
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$F = kx = 200 \times 0.10 = 20\ \text{N}$.
A force of $15\ \text{N}$ stretches a spring by $0.05\ \text{m}$. What is the spring constant, in $\tfrac{\text{N}}{\text{m}}$?
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$k = F/x = 15 / 0.05 = 300\ \tfrac{\text{N}}{\text{m}}$.
Within the elastic limit, doubling a spring's extension changes its force to:
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$F = kx$ is linear, so doubling $x$ doubles $F$.
The spring constant $k$ measures a spring's ____ (how much force each metre of stretch takes).
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A large $k$ means a stiff spring — more force per metre of extension.
Always a restoring force
- Whichever way you deform the spring, its force points back toward equilibrium.
- Stretch it and it pulls in; compress it and it pushes out.
- That is why we write $F = -kx$ — the minus sign means "opposite to the displacement".
- This restoring behaviour is exactly what makes springs oscillate (Topic 7).
A stretched spring pushes in the same direction as the stretch.
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It is a restoring force, pointing back toward equilibrium — opposite to the stretch ($F = -kx$).
The elastic limit
- Hooke's law only holds up to the spring's elastic limit 弹性极限.
- Stretch beyond it and the spring deforms permanently — the straight line bends.
- Within the limit, the spring returns to its original length when released.
- Every real spring has a range where $F = kx$ is a faithful description.
Select all statements that are true about Hooke's law.
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Hooke's law is linear, its slope is $k$, and the force restores. But it fails beyond the elastic limit.
The spring force is a restoring force: it points opposite to the stretch, not along it. And Hooke's law is only valid below the elastic limit — push a spring too far and $F = kx$ no longer holds.
A spring with $k = 200\ \tfrac{\text{N}}{\text{m}}$ is stretched by $x = 0.10\ \text{m}$.
- $F = kx = 200 \times 0.10 = 20\ \text{N}$.
- The spring pulls back with $20\ \text{N}$ toward its natural length.
Hooke's law: a spring's force is proportional to its extension, $F = kx$ (as a restoring force, $F = -kx$). The spring constant $k$ (in $\tfrac{\text{N}}{\text{m}}$) is its stiffness — the slope of the force–extension line. It holds only below the elastic limit.
Write from a spring constant of 30
- A 12 N force stretches a spring by 0.40 m.
- Find the spring constant and the direction of the spring force.
A force of 12 N stretches a spring by 0.40 m. What is the spring constant, in N/m, and which way does the spring force point?
12 divided by 0.40 is 30.
What is the spring constant?
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Force divided by extension is 12 divided by 0.40. Other options — 300 N/m (the lesson spring); 12 N/m (the force used as k).