# Question 4:

## Part 4(a):

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Drum **smooth**, or no friction (therefore reaction is perpendicular to the ramp) | B1 | AO 2.4 |

## Part 4(b):

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $(\nearrow):\ F\cos\theta + R\sin\theta = 20g\sin\theta$ | M1, A1 | N.B. In moments equation, if there is an extra $\sin\theta$ or $\cos\theta$ on a length, give M0 for that equation |
| $(\nwarrow):\ N + R\cos\theta = 20g\cos\theta + F\sin\theta$ | M1, A1 | |
| $(\uparrow):\ R + N\cos\theta = 20g$ | | |
| $(\rightarrow):\ F = N\sin\theta$ | | |
| $M(A):\ 20g \times 4\cos\theta = 5N$ | M1, A1 | |
| $M(B):\ 3N + R\times8\cos\theta = F\times8\sin\theta + 20g\times4\cos\theta$ | | |
| $M(C):\ R\times5\cos\theta = F\times5\sin\theta + 20g\times\cos\theta$ | | |
| $M(G):\ R\times4\cos\theta = F\times4\sin\theta + N$ | A1 | Values: $N=150.528;\ F=42.14784;\ R=51.49312$ |
| **Alternative 1** (components along ramp $X$, perp to ramp $Y$): | | |
| $(\nearrow):\ X = 20g\sin\theta$ | M1, A1 | |
| $(\nwarrow):\ Y + N = 20g\cos\theta$ | M1, A1 | |
| $M(A):\ 20g\times4\cos\theta = 5N$ | M1, A1 | Values: $N=150.528;\ X=54.88;\ Y=37.632$ |

# Question 4 (Alternative 2):

**Along ramp ($\nearrow$):** $H\cos\theta = 20g\sin\theta$ | M1 | A1 | 3.3, 1.1b

**Perpendicular to ramp ($\nwarrow$):** $S + N = H\sin\theta + 20g\cos\theta$ | A1 | 1.1b

**Vertical ($\uparrow$):** $S\cos\theta + N\cos\theta = 20g$ | M1 | 3.4

**Horizontal ($\rightarrow$):** $H = S\sin\theta + N\sin\theta$ | — | —

**M(A):** $20g \times 4\cos\theta = 5N$ | A1 | 1.1b

**M(B):** $20g \times 4\cos\theta + H \times 8\sin\theta = 8S + 3N$ | M1 | 3.4

**M(C):** $20g \times \cos\theta + H \times 5\sin\theta = 5S$ | — | —

**M(G):** $4S = N \times 1 + H \times 4\sin\theta$ | A1 | 1.1b

Values: $N = 150.528$; $H = 57.1666\ldots$; $S = 53.638666\ldots$ | — | —

Solve 3 equations for $F$ and $R$ **OR** $X$ and $Y$ **OR** $H$ and $S$ | M1 | 1.1b

$|\text{Force}| = \sqrt{R^2 + F^2}$ (Main scheme) **OR** $= \sqrt{X^2 + Y^2}$ (Alt 1) **OR** $= \sqrt{H^2 + S^2 - 2HS\cos(90° - \theta)}$ (Alt 2) | M1 | 3.1b

Magnitude $= 67$ or $66.5$ (N) | A1 | 2.2a

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# Question 4(c):

Magnitude of the normal reaction (at $C$) will **decrease** | B1 | 3.5a

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**Examiner Notes for Q4:**

- 4a: B1 | Ignore any extra incorrect comments
- 4b: Generally 3 independent equations required so **at least one moments equation**: M1A1M1A1M1A1. More than 3 equations, give marks for best 3. M1 requires all terms, dimensionally correct (condone sin/cos confusion). A1 for correct unsimplified equation. If reaction at $C$ is not perpendicular to ramp, can only score marks for M(C). Allow use of $(\mu R)$ for $F$
- Alt 1/Alt 2: Same M1A1 criteria apply; N.B. can find $H$ and $S$ using only TWO equations (1st and 7th in list); mark better equation as M2A2, second as M1A1
- Final M1: Substitute trig and solve for two components — must use 3 equations unless special case where 2 is sufficient
- Final M1: Pythagoras to find magnitude (independent mark but must have found values for both components) OR cosine rule — correct answer only for A1

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