## Question 5:

**Part 5a:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Moments about $A$ | M1 | Dimensionally correct equation i.e. force $\times$ distance $=$ force $\times$ distance. Condone sin/cos confusion. Mark $50g$ as an accuracy error |
| $4T = 2\cos\alpha \times 50 \quad \left(= 2 \times \frac{4}{5} \times 50\right)$ | A1 | Correct unsimplified equation. Need to see $\cos\alpha$ OR $\frac{4}{5}$. Might see LHS $= T\cos\alpha \times 4\cos\alpha + T\sin\alpha \times 4\sin\alpha$ |
| $T = 20\ *$ | A1* | Obtain given answer from correct working. Must see $\frac{4}{5}$ used correctly |
| **Total [3]** | | |

**Part 5b:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Resolve horizontally | M1 | Condone sin/cos confusion |
| $H = T\sin\alpha$ | A1 | Correct equation |
| Resolve vertically | M1 | Need all 3 terms. Condone sign error and sin/cos confusion |
| $T\cos\alpha + V = 50$ | A1 | Correct equation |
| Either or both of the above equations could be replaced by a moments equation e.g. $M(B): 4\cos\alpha \times V = 4\sin\alpha \times H + 2\cos\alpha \times 50$ or by resolving perpendicular & parallel to the rod: $T + V\cos\alpha = 50\cos\alpha + H\sin\alpha$ & $50\sin\alpha = H\cos\alpha + V\sin\alpha$ | | |
| Use $F = \mu R$ to form an equation in $\mu$ | M1 | $(H = \mu V)$ Used, not just stated i.e. they must get as far as substituting their values |
| $\mu = \frac{6}{17}$ | A1 | $\mu = 0.35$ or better. Accept $\frac{12}{34}$ |
| **Total [6]** | | |

## Question 6a:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of $I = mv - mu$ for $P$ or $Q$ | M1 | Dimensionally correct. Need all terms. M0 if $m$ is missing on RHS |
| $5mv = m(2v-(-y))$ or $-5mv = km(v-x)$ | A1 | Correct unsimplified equation |
| Use of CLM or second use of $I = mv - mu$ | M1 | Dimensionally correct. Need all terms. In CLM allow cancelled $m$ and extra common factor (e.g. $g$) throughout |
| $kmx - my = kmv + 2mv$ or $(kx - y = kv + 2v)$ or $-5mv = km(v-x)$ | A1 | Correct unsimplified equation |
| Use of impact law | M1 | Must be used with $e$ on the correct side. Condone sign errors |
| $2v - v = \frac{1}{5}(x+y)$ | A1 | Correct unsimplified equation |
| $y = 3v$ | A1 | cao |
| $x = 2v$ | A1 | cao |
| $k = 5$ | A1 | cao |
| **[9]** | | |

## Question 6b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| KE lost | M1 | Dimensionally correct. Accept change in KE. Not scored until they form the complete substituted equation |
| $= \frac{1}{2} \times km(x^2 - v^2) + \frac{1}{2} \times m(y^2 - 4v^2)$ $\left(= \frac{15}{2}mv^2 + \frac{5}{2}mv^2\right)$ | A1ft | Correct unsimplified expression. Follow their $x$, $y$, $k$. Condone sign change without explanation. KE before $= 14.5mv^2$, KE after $= 4.5mv^2$ |
| $= 10mv^2$ | A1 | Only |
| **[3]** | | |

---

## Question 7a:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Mass ratios: $PQUV = 9a^2$, $URST = 36a^2$, $QRU = 18a^2$, total $= 63a^2$ | B1 | Correct mass ratios (1:4:2:7) |
| Displacements from $QT$: $-\frac{3a}{2}$, $3a$, $2a$, $d$ | B1 | Correct displacements from $QT$ or a parallel axis seen or implied. Signs consistent |
| **Equation** for moments about $QT$ | M1 | (Or a parallel axis) Dimensionally correct. Condone sign errors |
| $18 \times 2a + 36 \times 3a - 9 \times \frac{3a}{2} = 63d$ $\left(4a + 12a - \frac{3a}{2} = 7d\right)$ | A1 | Or equivalent. Correct unsimplified equation. Check consistent in $a$ |
| $d = \frac{29a}{2} \div 7 \left(= \frac{261a}{2} \div 63\right) = \frac{29a}{14}$ | A1* | Obtain **given answer** from correct working. Need to see at least one interim step with all the $a$ terms collected. Check $a$ is in final answer |
| **[5]** | | |

## Question 7b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Vertical distances from $Q$: $\frac{3a}{2}$, $6a(= 3a+3a)$, $2a$, $(v)$. From $T$: $7.5a$, $3a$, $7a$ | B1 | Seen or implied |
| **Equation** for moments about $PQ$ | M1 | (Or a parallel axis) Dimensionally correct. Condone sign errors |
| $9 \times \frac{3a}{2} + 18 \times 2a + 36 \times 6a = 63v$ $\left(\frac{3a}{2} + 2 \times 2a + 4 \times 6a = 7v\right)$ | A1 | Correct unsimplified equation |
| $v = \frac{59a}{14}$ $\left(\frac{67}{14}a \text{ above } T, \frac{17}{14}a \text{ below } U\right)$ | A1 | $4.2a$ or better $(4.214\ldots)$ |
| $\tan\alpha = \frac{29}{59}$ $(= 26.175\ldots°)$ | M1 | Use trig and their $v$ to find a relevant angle. Allow for $90° - 26.17\ldots°$ |
| $\theta° = \tan^{-1}2 - \tan^{-1}\left(\frac{29}{59}\right)$ | M1 | Use their $v$ to find the required angle $(63.43\ldots° - 26.175\ldots°)$ |
| $\theta = 37.3$ | A1 | 37 or better |
| **[7]** | | |

---

## Question 8a:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Normal reaction between $P$ and the ramp $= 3g\cos\alpha$ $\left(= \frac{18g}{\sqrt{37}} = 29.0\right)$ | B1 | cao ISW |
| Use of $F = \frac{3}{4}R$ | M1 | $\frac{3}{4} \times$ their $R$ (Must have an $R$) |
| Work done $= 4F$ | M1 | Their $F$ (Must have an $F$) |
| $= 87.0\ (87)\ \text{(J)}$ | A1 | 3 sf or 2 sf only (follows 9.8). Do not allow $\frac{54}{\sqrt{37}}g$ (this is an acceleration) |
| **[4]** | | |

## Question 8b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Work-energy equation | M1 | **M0 if not using work-energy**. All terms required. Condone sign errors. Condone sin/cos confusion |
| $\frac{1}{2} \times 3U^2 - \text{their(a)} - 3g \times 4\sin\alpha = \frac{1}{2} \times 3 \times 25$ | A1ft | Unsimplified equation with at most one error. Follow their (a) |
| (correct unsimplified equation) | A1ft | Correct unsimplified equation. Follow their (a) |
| $U = 9.79$ or $U = 9.8$ | A1 | 3 sf or 2 sf only (follows 9.8) |
| **[4]** | | |

## Question 8c:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Time taken: | M1 | Complete method using suvat and $u = 5$ to form an equation in $t$ only |
| $-4\sin\alpha = (5\sin\alpha)t - \frac{1}{2}gt^2$ $\left(4.9\sqrt{37}t^2 - 5t - 4 = 0\right)$ | A1 | Correct unsimplified equation for $t$ |
| $t = 0.45969\ldots$ | A1 | Seen or implied |
| Horizontal distance | M1 | Complete method using suvat and $u = 5$ |
| $= (5\cos\alpha)t$ $\left(= \frac{30}{\sqrt{37}}t\right)$ | A1ft | Follow their $t$ |
| $= 2.27$ or $2.3\ \text{(m)}$ | A1 | 3 sf or 2 sf only |
| **[6]** | | |