# Question 7:

## Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Conservation of energy: $\frac{1}{2}mu^2 = \frac{1}{2}mv^2 + mg \times \frac{2a}{5}(1-\cos\theta)$ | M1, A1 | Need all terms, dimensionally correct. Condone sign errors and sin/cos confusion. Allow with $\frac{2a}{5}\cos\theta$ in place of $\frac{2a}{5}(1-\cos\theta)$ |
| Equation of motion towards $O$: $T - mg\cos\theta = \frac{5mv^2}{2a}$ | M1, A1 | Need all terms, dimensionally correct. Condone sign errors and sin/cos confusion |
| Complete method to find $T$ in terms of $u$, $a$ and $\theta$ | DM1 | Complete method using conservation of energy and circular motion to eliminate $v$. Requires two preceding M marks |
| $T = mg\cos\theta + \frac{5m}{2a}\left(u^2 - \frac{4a}{5}g(1-\cos\theta)\right) = 3mg\cos\theta - 2mg + \frac{5mu^2}{2a}$ | A1* | Obtain given result from correct working |

## Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Require $T \geq 0$ when $\theta = \pi$: $\frac{5mu^2}{2a} \geq mg(2+3)$ | M1 | Identify correct condition for complete circle, solve for $u$. Condone working from $T=0$ |
| $u^2 \geq 2ag$, minimum $u = \sqrt{2ag}$ | A1 | Allow $u \geq \sqrt{2ag}$. Condone $u > \sqrt{2ag}$ and $u = \sqrt{2ag}$ |

## Part (c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\theta = \frac{\pi}{2}, u = 2\sqrt{ag} \Rightarrow T = -2mg + \frac{5m}{2a}\times 4ag = 8mg$ | B1 | Correct $T$ or $v^2$ seen or implied |
| Magnitude of acceleration $= g\sqrt{64+1} = \sqrt{65}g$ | M1, A1 | Use Pythagoras with horizontal component of acceleration. Correct only, or $8.1g$ ($8.062\ldots g$) or better |

## Part (d):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Consider the uniformity/dimensions of the package; string might be extensible; include the weight of the string | B1 | Any valid suggestion **relating to the model**. Allow negatives of statements within the model. B0 if multiple suggestions including one incorrect. B0 for accuracy of $g$ |

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