# Question 6 (Alternative - combining parts):

| Working | Mark | Notes |
|---------|------|-------|
| Moments about $C$: $20 \times 0.5\cos\theta + F \times 2.5\sin\theta = R \times 2.5\cos\theta$ | M1 | From (b). Dimensionally correct. Condone sin/cos confusion and sign errors |
| Use $F = \frac{1}{4}R$ | B1 | From (b) |
| $R = \frac{64}{13}$, $(R = 4.92...)$ | A1 | From (b) |
| Resolve perpendicular to rod: $N + R\cos\theta = \mu R\sin\theta + W\cos\theta$ | M1 | *From (a) |
| $N + R\cos\theta = \mu R\sin\theta + W\cos\theta$ | A1 | From (a). Correct unsimplified equation |
| $N = 12.8$ (N) | A1 | From (a) |
| Resolve parallel rod: $P + F\cos\theta + R\sin\theta = 20\sin\theta \, (= 12)$ | M1 A1 | Correct unsimplified equation |
| $P = 12 - \frac{R}{4} \times \frac{4}{5} - R \times \frac{3}{5} = 12 - \frac{4}{5}R = 8.06...$ | DM1 | Dependent on preceding 2 M marks |
| Solve for $\mu$: $P = \mu N$, $\mu = 0.630$ (0.63) | DM1 A1 | *Could use an alternative pair of resolutions |

**Total: [11]**

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# Question 7a:

| Working | Mark | Notes |
|---------|------|-------|
| CLM (taking initial direction of $P$ to be +ve): $3m \times u - 2m \times 2u = 3mv + 2mw \, (= -mu)$ | M1 A1 | All terms needed. Dimensionally correct. Condone sign error(s). Correct unsimplified equation |
| Negative total $\Rightarrow v < 0$ and $w < 0$, or $v < 0$ and $w > 0$; because $w < 0$ and $v > 0$ is impossible, therefore $v < 0$ — $P$ has changed direction. **Given answer** | A1 | Alt: solve CLM & impact equation for $v\left(= -\frac{u}{5}(1+6e)\right)$ and use range of possible values for $e$ to justify given answer |

**Total: (3)**

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# Question 7b:

| Working | Mark | Notes |
|---------|------|-------|
| Impact law: $w - v = e(u + 2u) = 3eu$ | M1 A1 | Used correctly. Correct unsimplified equation. Signs consistent with CLM equation. Allow M1A1 if seen in (a) and used in (b) |
| Solve for $kw$: $v = w - 3eu \Rightarrow 3(w - 3eu) + 2w = -u$, $5w = 9eu - u$ | DM1 | Dependent on preceding M1 |
| Correct inequality for their $w$: $w > 0 \Rightarrow 9eu - u > 0$ | DM1 | Form inequality in $e$. Dependent on preceding M1 |
| $(1 \geq) \; e > \frac{1}{9}$ | A1 | Allow if upper limit not stated. A0 if upper limit incorrect. Condone $1 > e$ |

**Total: (5)**

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# Question 7c:

| Working | Mark | Notes |
|---------|------|-------|
| $v$ and $w$ in terms of $u$: $5w = \frac{9}{2}u - u = \frac{7}{2}u$, $w = \frac{7}{10}u$ | M1 | Solve for $v$ or $w$ |
| $v = w - \frac{3}{2}u = -\frac{8}{10}u$ | A1 | Both values correct |
| Loss in KE | M1 | Accept change in KE. Must be using two different masses ($3m$ and $2m$) |
| $= \frac{1}{2} \times 3m \times (u^2 - v^2) + \frac{1}{2} \times 2m \times (4u^2 - w^2)$ | A1ft | Follow their $v$, $w$. Correct unsimplified equation |
| $= \frac{3}{2}mu^2(1 - 0.64) + mu^2(4 - 0.49) = 4.05mu^2$ | A1 | $\left(\frac{81}{20}mu^2 \text{ or equivalent}\right)$ |

**Total: (5) [13]**

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# Question 8a:

| Working | Mark | Notes |
|---------|------|-------|
| Change in KE: $\frac{1}{2} \times 3 \times (15^2 - 10^2) \, (= 187.5)$ (J) | B1 | One term correct unsimplified |
| Gain in GPE: $3g \times 6\sin 20 \, (= 60.3...)$ (J) | B1 | Two terms correct unsimplified |
| Work done against friction: $6F$ | B1 | All three terms correct unsimplified |
| Work-energy: $187.5 = 6F + 18g\sin 20$ | M1 A1 | Dimensionally correct. All terms needed. Condone sign errors and sin/cos confusion. Correct unsimplified equation |
| $F = 21.2$ (21) | A1 | |

**Total: (6)**

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# Question 8b:

| Working | Mark | Notes |
|---------|------|-------|
| Energy: $\frac{1}{2} \times 3 \times 100 + 3 \times 9.8 \times 6\sin 20 = \frac{1}{2} \times 3 \times w^2$ | M1 A1 | Dimensionally correct. All terms needed. Condone sign errors and sin/cos confusion. Correct unsimplified equation |
| $w = 11.8$ (12) (m s$^{-1}$) | A1 | |
| Direction: $\cos\alpha = \frac{10\cos 20}{11.84...}$ | M1 | Use trig to find a relevant angle |
| $37.5°$ ($37°$) below the horizontal | A1 | |

**Total: (5)**

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# Question 8b (Alternative):

| Working | Mark | Notes |
|---------|------|-------|
| Find components and use Pythagoras: $w = \sqrt{(10\sin 20)^2 + 12g\sin 20 + (10\cos 20)^2}$ | M1 A1 | Condone sign errors and sin/cos confusion. Correct unsimplified equation $(v_x = 9.396..., \; v_y = 7.205...)$ |
| $w = 11.8$ (12) (m s$^{-1}$) | A1 | |
| Direction: $\tan\alpha = \frac{\sqrt{(10\sin 20)^2 + 12g\sin 20}}{10\cos 20}$ | M1 | Use trig to find a relevant angle |
| $37.5°$ ($37°$) below the horizontal | A1 | |

**Total: (5)**

## Question 8c:

**Method 1 (suvat):**

Use suvat to find height above $B$ | M1 | Complete method

$(10\sin 20)^2 = 2g \times s \quad (s = 0.5968....)$ | A1 | Correct unsimplified equation in $s$

Total height $= s + 6\sin 20$ | DM1 | Dependent on the preceding M1

$= 2.65\ (2.6)$ (m) | A1 |

**(4)**

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**Method 2 (alt - vertical component):**

Vertical component of $w$ (by trig or Pythagoras) | M1 |

$= 7.205..$ | A1 |

Max ht $= \dfrac{v^2}{2g}$ | DM1 | Dependent on the preceding M1

$= \dfrac{7.205^2}{19.6} = 2.65$ | A1 |

**(4)**

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**Method 3 (alt - conservation of energy):**

Conservation of energy: | M1 | Using speed at max ht $= 10\cos 20°$, need all terms, condone sign error

$\dfrac{1}{2} \times 3 \times (10\cos 20°)^2 + 3gh = \dfrac{1}{2} \times 3w^2$ | A1 | Correct unsimplified equation

$h = \dfrac{w^2 - 100\cos^2 20°}{2g}$ | DM1 | Substitute for $w$ and solve for $h$, dependent on the preceding M1

$= 2.65\ (2.6)$ (m) | A1 |

**(4)**

**[15]**