# Question 7:

## Part 7(a):

| Working/Answer | Marks | Guidance |
|---|---|---|
| $3mg\cdot 2l\sin\theta+4mg\cdot 4l\sin\theta=-2m(a^2+40l^2)\ddot{\theta}$ | M1 A1 | M1 for equation of motion about $A$ with usual rules; A1 for correct equation |
| For small $\theta$, $\sin\theta\approx\theta$: $-\frac{11gl}{(a^2+40l^2)}\theta=\ddot{\theta}$ | M1 | M1 for using small angle approx. and putting into appropriate form (with $-$ sign) |
| SHM; $T=2\pi\sqrt{\dfrac{(a^2+40l^2)}{11gl}}$ | M1 A1 | M1 for SHM and use of correct formula for period; A1 for correct answer |

## Part 7(b):

| Working/Answer | Marks | Guidance |
|---|---|---|
| $4mg\cdot 4l+3mg\cdot 2l=\frac{1}{2}\cdot 2m(a^2+40l^2)\omega^2$ | M1 A1 | M1 for conservation of energy equation with usual rules; A1 for correct equation |
| $\omega^2=\dfrac{22gl}{(a^2+40l^2)}$ | A1 | A1 for correct GIVEN ANSWER |

## Part 7(c):

| Working/Answer | Marks | Guidance |
|---|---|---|
| $J\cdot 2l=2m(a^2+40l^2)\!\left(\tfrac{1}{2}\omega-(-\omega)\right)$ | M1 A2 | M1A2 for angular impulse-momentum equation |
| $J=\dfrac{3m}{2}\sqrt{\dfrac{22g(a^2+40l^2)}{l}}$ | A1 | A1 for correct answer |

## Part 7(d):

| Working/Answer | Marks | Guidance |
|---|---|---|
| $\frac{1}{2}\cdot 2m(a^2+40l^2)\!\left(\tfrac{1}{2}\omega\right)^2=3mg\cdot 2l\sin\theta+4mg\cdot 4l\sin\theta$ | M1 A1 | M1A1 for energy equation |
| $\frac{1}{8}\cdot 2m(a^2+40l^2)\cdot\dfrac{22gl}{(a^2+40l^2)}=22mgl\sin\theta$ | DM1 | DM1 for substituting $\omega^2$ |
| $\sin\theta=\dfrac{1}{4}\Rightarrow\theta=\sin^{-1}\!\dfrac{1}{4}$ | A1 | A1 for GIVEN ANSWER correctly obtained |

## Question 7:

### Part 7(c):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Impulse-momentum equation (general form, $I$ and $\omega$ not yet substituted) | M1 | $I$ and $\omega$ do not need to be substituted at this stage |
| Correct equation without $I$ and $\omega$ substituted | A1 | First correct equation |
| Correct equation with $I$ and $\omega$ substituted | A1 | Second correct equation |
| Correct final answer (must be positive) | A1 | Third mark |

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### Part 7(d):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Conservation of energy equation (general form) | M1 | First M1, usual rules apply |
| Correct equation without $\omega$ substituted | A1 | First A1 |
| Substituting for $\omega$ | DM1 | Dependent on first M1 |
| Correctly obtaining the given answer | A1 | Second A1, must arrive at GIVEN ANSWER |