A2 October 2020 Q6
6. A light elastic string with natural length \(l\) and modulus of elasticity \(kmg\) has one end attached to a fixed point \(A\) on a rough inclined plane. The other end of the string is attached to a package of mass \(m\).
The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan\theta = \dfrac{5}{12}\)
The package is initially held at \(A\). The package is then projected with speed \(\sqrt{6gl}\) up a line of greatest slope of the plane and first comes to rest at the point \(B\), where \(AB = 3l\).
The coefficient of friction between the package and the plane is \(\dfrac{1}{4}\)
By modelling the package as a particle,
| Scheme | Marks | AO |
|---|---|---|
| Work done against friction \(= 3l \times \mu mg\cos\theta\ \ \left(= \dfrac{9mgl}{13}\right)\) Gain in EPE \(= \dfrac{kmg \times 4l^2}{2l}\ \ (= 2kmgl)\) Gain in GPE \(= mg \times 3l\sin\theta\ \ \left(= \dfrac{15mgl}{13}\right)\) | B1 B1 B1 | 3.4 3.4 3.4 |
| Work energy equation: | M1 | 2.1 |
| \(\dfrac{1}{2}m \times 6gl = \dfrac{9mgl}{13} + 2kmgl + \dfrac{15mgl}{13}\) | A1 | 1.1b |
| \(2k = 3 - \dfrac{24}{13} = \dfrac{15}{13}\), \(k = \dfrac{15}{26}\) * | A1* | 2.2a |
| (6) |
Notes
B1 B1 B1: Use model to obtain one correct term
Use model to obtain two correct terms
Use model to obtain three correct terms
M1: Work-energy equation. Need all terms and no extras. Dimensionally correct. Condone sign errors and sin/cos confusion.
A1: Correct unsimplified equation
A1*: Obtain given result from correct working
NB: The use of suvat equations is not a valid alternative method because the acceleration is not constant
| Scheme | Marks | AO |
|---|---|---|
| Tension in the string at \(B\): \(\dfrac{\frac{15}{26}mg \times 2l}{l}\ \ \left(= \dfrac{15mg}{13}\right)\) | B1 | 3.1a |
| Equation of motion: tension + component of weight − friction \(= ma\) | M1 | 3.3 |
| \(\dfrac{15mg}{13} + mg\sin\theta - \dfrac{1}{4}mg\cos\theta = ma\) \(\left(mg\left(\dfrac{15}{13} + \dfrac{5}{13} - \dfrac{3}{13}\right) = ma\right)\) | A1 A1 | 1.1b 1.1b |
| \(a = \dfrac{17g}{13}\) | A1 | 1.1b |
| (5) | ||
| (11 marks) |
Notes
B1: Correct unsimplified expression for the tension in the string
M1: Equation of motion. Need all terms and no extras. Condone sign errors and sin/cos confusion Allow with \(T\) or their \(T\)
A1 A1: Unsimplified equation with at most one error
Correct unsimplified equation
A1: Exact answer or accept 12.8 or 13 \((\text{m s}^{-2})\)