A2 June 2022 Q7
7.
A package \(P\) of mass \(m\) is attached to one end of a string of length \(\dfrac{2a}{5}\). The other end of the string is attached to a fixed point \(O\). The package hangs at rest vertically below \(O\) with the string taut and is then projected horizontally with speed \(u\), as shown in Figure 5.
When \(OP\) has turned through an angle \(\theta\) and the string is still taut, the tension in the string is \(T\)
The package is modelled as a particle and the string as being light and inextensible.
Given that \(P\) moves in a complete vertical circle with centre \(O\)
Given that \(u = 2\sqrt{ag}\)
| Scheme | Marks | AO |
|---|---|---|
| Conservation of energy: | M1 | 3.1b |
| \(\dfrac{1}{2}mu^2 = \dfrac{1}{2}mv^2 + mg\times\dfrac{2a}{5}(1 - \cos\theta)\) | A1 | 1.1b |
| Equation of motion towards \(O\) | M1 | 3.1b |
| \(T - mg\cos\theta = \dfrac{5mv^2}{2a}\) | A1 | 1.1b |
| Complete method to find \(T\) in terms of \(u\), \(a\) and \(\theta\) | DM1 | 2.1 |
| \(T = mg\cos\theta + \dfrac{5m}{2a}\left(u^2 - \dfrac{4a}{5}g(1 - \cos\theta)\right)\) \(\phantom{T} = 3mg\cos\theta - 2mg + \dfrac{5mu^2}{2a}\) * | A1* | 2.2a |
| (6) |
Notes
M1: Need all terms. Dimensionally correct. Condone sign errors and sin/cos confusion
Allow with \(\dfrac{2a}{5}\cos\theta\) in place of \(\dfrac{2a}{5}(1 - \cos\theta)\)
A1: Correct unsimplified equation
M1: Need all terms. Dimensionally correct. Condone sign errors and sin/cos confusion
A1: Correct unsimplified equation
M1: Complete method, e.g. using conservation of energy and the circular motion, to form sufficient equations to obtain an expression without \(v\)
A complete method requires the two preceding M marks.
A1*: Obtain given result from correct working
| Scheme | Marks | AO |
|---|---|---|
| Require \(T \geqslant 0\) when \(\theta = \pi\): \(\dfrac{5mu^2}{2a} \geqslant mg(2 + 3)\) | M1 | 2.1 |
| \(u^2 \geqslant 2ag,\quad \text{minimum } u = \sqrt{2ag}\) | A1 | 1.1b |
| (2) |
Notes
M1: Identify correct condition for complete circle and solve for \(u\). Condone working from \(T = 0\)
A1: Allow \(u \geqslant \sqrt{2ag}\)
Condone \(u \gt \sqrt{2ag}\), and \(u = \sqrt{2ag}\)
| Scheme | Marks | AO |
|---|---|---|
| \(\theta = \dfrac{\pi}{2},\ u = 2\sqrt{ag} \ \Rightarrow\ T = -2mg + \dfrac{5m}{2a}\times 4ag\) | B1 | 1.1b |
 | ||
| Magnitude of acceleration \(= g\sqrt{64 + 1}\) | M1 | 2.1 |
| \(= \sqrt{65}g\) | A1 | 1.1b |
| (3) |
Notes
B1: Correct \(T\) or \(v^2\) seen or implied
M1: Use of Pythagoras with their horizontal component of acceleration
A1: Correct only, or \(8.1g\) \((8.062\ldots g)\) or better
| Scheme | Marks | AO |
|---|---|---|
| Consider the uniformity / dimensions of the package String might be extensible. include the weight of the string | B1 | 3.5c |
| (1) | ||
| (12 marks) |
Notes
B1: Any valid suggestion relating to the model.
Allow negatives of statements within the model
e.g. not model the package as a particle.
B0 if multiple suggestions including one incorrect.
B0 for accuracy of \(g\) as this is not part of the description of the model.