Standard +0.8 This is a multi-step energy problem requiring careful consideration of elastic potential energy, work done against friction, and the changing direction of friction. Students must recognize that the spring is initially compressed, then extends beyond natural length, requiring them to split the motion into phases and apply energy conservation with friction correctly in each phase. The modulus being expressed as 2mg adds algebraic complexity, and determining when the particle stops requires solving a non-trivial equation.
2. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring, of natural length \(l\) and modulus of elasticity \(2 m g\). The other end of the spring is attached to a fixed point \(A\) on a rough horizontal plane. The particle is held at rest on the plane at a point \(B\), where \(A B = \frac { 1 } { 2 } l\), and released from rest. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\)
Find the distance of \(P\) from \(B\) when \(P\) first comes to rest.
M1: difference of 2 elastic energy terms, not necessarily in complete energy equation. A1: correct difference. Second M1: work-energy equation, loss of EPE = work done against friction (not dep on previous mark). A1: fully correct equation
\(8x^2 + 2lx - l^2 = 0\)
M1 A1
M1dep: re-arranging to three term quadratic, dependent on second M mark, or use difference of 2 squares to get linear equation. A1: correct 3 term quadratic, terms in any order
\((4x-l)(2x+l) = 0\)
M1dep
M1dep: solving the resulting quadratic, usual rules. Dependent on all second and third M marks
## Question 2:
| Working/Answer | Marks | Guidance |
|---|---|---|
| $\frac{2mg}{2l}\left(\left(\frac{1}{2}l\right)^2 - x^2\right) = \frac{1}{4}mg\left(\frac{1}{2}l + x\right)$ | M1 A1; M1 A1 | M1: difference of 2 elastic energy terms, not necessarily in complete energy equation. A1: correct difference. Second M1: work-energy equation, loss of EPE = work done against friction (not dep on previous mark). A1: fully correct equation |
| $8x^2 + 2lx - l^2 = 0$ | M1 A1 | M1dep: re-arranging to three term quadratic, dependent on second M mark, or use difference of 2 squares to get linear equation. A1: correct 3 term quadratic, terms in any order |
| $(4x-l)(2x+l) = 0$ | M1dep | M1dep: solving the resulting quadratic, usual rules. Dependent on all second and third M marks |
| $x = \frac{1}{4}l$ or $-\frac{1}{2}l$ | A1 | $x = -\frac{1}{2}l$ need not be shown |
| distance $= \frac{1}{2}l + \frac{1}{4}l = \frac{3}{4}l$ | A1 | A1cao and cso |
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2. A particle $P$ of mass $m$ is attached to one end of a light elastic spring, of natural length $l$ and modulus of elasticity $2 m g$. The other end of the spring is attached to a fixed point $A$ on a rough horizontal plane. The particle is held at rest on the plane at a point $B$, where $A B = \frac { 1 } { 2 } l$, and released from rest. The coefficient of friction between $P$ and the plane is $\frac { 1 } { 4 }$
Find the distance of $P$ from $B$ when $P$ first comes to rest.\\
\hfill \mbox{\textit{Edexcel M3 2014 Q2 [9]}}