| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Elastic string on rough inclined plane |
| Difficulty | Challenging +1.2 This is a standard M3 energy conservation problem with elastic strings on an inclined plane. While it requires careful bookkeeping of multiple energy terms (gravitational PE, elastic PE, work against friction) and solving a quadratic equation, the setup is routine for this module with no novel insights needed. The multi-step nature and algebraic manipulation place it slightly above average difficulty. |
| Spec | 3.03v Motion on rough surface: including inclined planes6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Elastic energy \(= \dfrac{1}{2} \times 2mg \dfrac{x^2}{l}\) | B1 | Correct elastic potential energy |
| Work done by friction \(= (l+x)\mu mg\cos\alpha\) | B1 | Correct work done by friction |
| \((l+x)\mu mg\cos\alpha + \dfrac{1}{2}\times 2mg\dfrac{x^2}{l} = (l+x)mg\sin\alpha\) | M1A1ft | Attempt work-energy equation; must have 3 terms: friction, elastic energy, GPE; EPE term of form \(k\lambda\dfrac{x^2}{l}\); work done term of form distance \(\times \mu mg\cos\) or \(\sin\alpha\) |
| \(\dfrac{1}{4}\times\dfrac{4}{5}(l+x) + \dfrac{x^2}{l} = \dfrac{3}{5}(l+x)\) | ||
| \(5x^2 - 2lx - 2l^2 = 0\) | ||
| \(x = 0.863\ldots l\) | dM1A1 | Solve 3-term quadratic; correct extension decimal or exact; positive root only |
| \(k = 1.86\) | A1 [7] | Complete by adding 1 to numerical multiple of \(l\); must be 3 significant figures |
# Question 4:
| Working/Answer | Marks | Guidance |
|---|---|---|
| Elastic energy $= \dfrac{1}{2} \times 2mg \dfrac{x^2}{l}$ | B1 | Correct elastic potential energy |
| Work done by friction $= (l+x)\mu mg\cos\alpha$ | B1 | Correct work done by friction |
| $(l+x)\mu mg\cos\alpha + \dfrac{1}{2}\times 2mg\dfrac{x^2}{l} = (l+x)mg\sin\alpha$ | M1A1ft | Attempt work-energy equation; must have 3 terms: friction, elastic energy, GPE; EPE term of form $k\lambda\dfrac{x^2}{l}$; work done term of form distance $\times \mu mg\cos$ or $\sin\alpha$ |
| $\dfrac{1}{4}\times\dfrac{4}{5}(l+x) + \dfrac{x^2}{l} = \dfrac{3}{5}(l+x)$ | | |
| $5x^2 - 2lx - 2l^2 = 0$ | | |
| $x = 0.863\ldots l$ | dM1A1 | Solve 3-term quadratic; correct extension decimal or exact; positive root only |
| $k = 1.86$ | A1 [7] | Complete by adding 1 to numerical multiple of $l$; must be 3 significant figures |4. One end of a light elastic string, of modulus of elasticity $2 m g$ and natural length $l$, is fixed to a point $O$ on a rough plane. The plane is inclined at angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 3 } { 5 }$. The other end of the string is attached to a particle $P$ of mass $m$ which is held at rest on the plane at the point $O$. The coefficient of friction between $P$ and the plane is $\frac { 1 } { 4 }$. The particle is released from rest and slides down the plane, coming to instantaneous rest at the point $A$, where $O A = k l$.
Given that $k > 1$, find, to 3 significant figures, the value of $k$.
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\hfill \mbox{\textit{Edexcel M3 2018 Q4 [7]}}