## Question 6:

### Part 6(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| GPE lost by $B$ – GPE gained by $A$ | M1 | Expression for change in GPE. Must be dimensionally correct and resolved terms where necessary. Allow subtraction either way round |
| $= 4 \times g \times 3 - 2 \times g \sin\theta \times 3$ | A1 | Correct unsimplified expression for change in PE (before substitution for $\sin\theta$). Allow subtraction either way round |
| $= 82\ (82.3)$ (J) | A1 | 2 sf or 3 sf. Accept $8.4g$ or $\frac{42g}{5}$. Must be positive but condone a sign change at end without explanation |

**(3 marks)**

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### Part 6(b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Total KE gained $= \frac{1}{2} \times 6 \times 4.5^2\ (= 60.75)$ (J) | B1 | Gain in KE for the system (not just one block) |
| Max friction $\mu 2g\cos\theta\ (= \mu \times 2 \times 9.8 \times \cos\theta = 15.68\mu)$ | B1 | Correct unsimplified expression for $F_{\max}$ seen or implied |
| Work done against friction $= 3 \times F_{\max}\ (= 47.04\mu)$ | B1ft | Correct expression for work done; follow their $F_{\max}$. Dependent on having found $F_{\max}$ |
| Work-energy equation: their GPE lost = their KE gained + their WD against friction | M1 | Complete method using work-energy to form equation in $\mu$. Require all terms (KE and GPE of both blocks). Dimensionally correct. Condone sign errors |
| $82.32 = 60.75 + 47.04\mu$ | A1 | Correct unsimplified equation in $\mu$ |
| $\mu = 0.459\ (0.46)$ | A1 | 3 sf or 2 sf only |

**(6 marks)**

> **NB:** Alternative via tension in string: B1 for KE of $B$ and correct tension (25.7 N); B1 for $F_{\max}$; B1ft for work done by tension and against friction; M1 for $3 \times 25.7 = 20.25 + 35.28 + 3 \times 15.68\mu$

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### Part 6(c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Work-energy equation for $A$ | M1 | All terms required. Dimensionally correct. Condone sign errors and $\sin/\cos$ confusion. If equation uses $d+3$ in place of $d$ in PE term, correct only if also includes term for initial PE. If $d+3$ used in work done term, scores M0 |
| $\frac{1}{2} \times 2 \times 4.5^2 = 2g\sin\theta \times d + 2g\cos\theta \times \mu d$ $\left(= 19.6 \times \frac{3}{5} \times d + 19.6 \times \frac{4}{5} \times \mu d\right)$ | A1ft | Unsimplified equation in $d$ and $\mu$ with at most one error |
| | A1ft | Correct unsimplified equation in $d$ and $\mu$. ft is on their $\mu$ if substituted |
| $d = 1.07\ (1.1)$ | A1 | 3 sf or 2 sf only |

**(4 marks)**

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