Challenging +1.2 Part (a) is a standard energy conservation problem requiring one equation. Part (b) requires setting up the condition for complete circular motion (tension ≥ 0 at highest point) and combining with energy conservation, which is a multi-step problem requiring insight into the constraint AB < l and the geometry of the new circular path. This is moderately above average for M3 but follows established circular motion patterns.
6. One end of a light inextensible string of length \(l\) is attached to a particle \(P\) of mass \(2 m\). The other end of the string is attached to a fixed point \(A\). The particle is hanging freely at rest with the string vertical. The particle is then projected horizontally with speed \(\sqrt { \frac { 7 g l } { 2 } }\) (a) Find the speed of \(P\) at the instant when the string is horizontal.
(4)
When the string is horizontal and \(P\) is moving upwards, the string comes into contact with a small smooth peg which is fixed at the point \(B\), where \(A B\) is horizontal and \(A B < l\). The particle then describes a complete semicircle with centre \(B\).
(b) Show that \(A B \geqslant \frac { 1 } { 2 } l\)
Use \(T \geqslant 0\) to obtain inequality for \(V^2\)
\(\frac{3gl - 4gr}{2r} \geqslant g\)
DM1
Use energy equation to eliminate \(V^2\); dependent on first and second M marks
\(\frac{3gl}{2} - 2gr \geqslant gr\)
\(r \leqslant \frac{1}{2}l\)
A1
Correct maximum value for \(r\)
\(AB \geqslant \frac{1}{2}l\) ✳
A1cso (9) [13]
Correct inequality for \(AB\); candidates using \(m\) in NL2 or assuming \(T=0\) cannot be awarded this mark
# Question 6:
## Part (a):
| Working | Marks | Guidance |
|---------|-------|----------|
| Energy to horizontal: $\frac{1}{2} \times 2m \times \frac{7gl}{2} - \frac{1}{2} \times 2mv^2 = 2mgl$ | M1A1A1 | Must be clear energy is being used; mass can be $m$ or $2m$; mixed masses are accuracy errors |
| $v = \sqrt{\frac{3gl}{2}}$ | A1 (4) | Correct speed at horizontal regardless of mass used |
## Part (b):
| Working | Marks | Guidance |
|---------|-------|----------|
| Energy from horizontal to top: $\frac{1}{2} \times 2m \times \frac{3gl}{2} - \frac{1}{2} \times 2mV^2 = 2mgr$ | M1A1 | Attempt energy equation from horizontal to top of new circle with unknown radius; mass can be $m$, $2m$ or mixed |
| $V^2 = \frac{3gl}{2} - 2gr$ | | |
| NL2 at top: $\frac{2mV^2}{r} = 2mg + T$ | M1A1A1 | NL2 at top of small circle; mass can be $m$, $2m$ or mixed; allow with $T=0$ |
| $T \geqslant 0 \Rightarrow \frac{2mV^2}{r} \geqslant 2mg$ | M1 | Use $T \geqslant 0$ to obtain inequality for $V^2$ |
| $\frac{3gl - 4gr}{2r} \geqslant g$ | DM1 | Use energy equation to eliminate $V^2$; dependent on first and second M marks |
| $\frac{3gl}{2} - 2gr \geqslant gr$ | | |
| $r \leqslant \frac{1}{2}l$ | A1 | Correct maximum value for $r$ |
| $AB \geqslant \frac{1}{2}l$ ✳ | A1cso (9) [13] | Correct inequality for $AB$; candidates using $m$ in NL2 or assuming $T=0$ cannot be awarded this mark |
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6. One end of a light inextensible string of length $l$ is attached to a particle $P$ of mass $2 m$. The other end of the string is attached to a fixed point $A$. The particle is hanging freely at rest with the string vertical. The particle is then projected horizontally with speed $\sqrt { \frac { 7 g l } { 2 } }$ (a) Find the speed of $P$ at the instant when the string is horizontal.\\
(4)
When the string is horizontal and $P$ is moving upwards, the string comes into contact with a small smooth peg which is fixed at the point $B$, where $A B$ is horizontal and $A B < l$. The particle then describes a complete semicircle with centre $B$.\\
(b) Show that $A B \geqslant \frac { 1 } { 2 } l$
\hfill \mbox{\textit{Edexcel M3 2016 Q6 [13]}}