A2 June 2022 Q6
6.
Two blocks, \(A\) and \(B\), of masses 2 kg and 4 kg respectively are attached to the ends of a light inextensible string.
Initially \(A\) is held on a fixed rough plane. The plane is inclined to horizontal ground at an angle \(\theta\), where \(\tan\theta = \dfrac{3}{4}\)
The string passes over a small smooth light pulley \(P\) that is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane.
Block \(A\) is held on the plane with the distance \(AP\) greater than 3 m.
Block \(B\) hangs freely below \(P\) at a distance of 3 m above the ground, as shown in Figure 4.
The coefficient of friction between \(A\) and the plane is \(\mu\)
Block \(A\) is released from rest with the string taut.
By modelling the blocks as particles,
Given that the speed of \(B\) at the instant it hits the ground is \(4.5\ \text{m s}^{-1}\) and ignoring air resistance,
After \(B\) hits the ground, \(A\) continues to move up the plane but does not reach the pulley in the subsequent motion.
Block \(A\) comes to instantaneous rest after moving a total distance of \((3 + d)\) m from its point of release.
Ignoring air resistance,
| Scheme | Marks | AO |
|---|---|---|
| GPE lost by \(B\) – GPE gained by \(A\) | M1 | 3.4 |
| \(= 4 \times g \times 3 - 2 \times g\sin\theta \times 3\) | A1 | 1.1b |
| \(= 82\ (82.3)\ (\text{J})\) | A1 | 1.1b |
| (3) |
Notes
M1: Expression for change in GPE. Must be dimensionally correct and resolved terms where necessary.
Allow subtraction either way round
A1: Correct unsimplified expression for the change in PE (before substitution for \(\sin\theta\))
Allow subtraction either way round
A1: 2 sf or 3 sf. Accept \(8.4g\) or \(\dfrac{42g}{5}\) ISW
Must be positive but condone a sign change at the end without explanation
| Scheme | Marks | AO |
|---|---|---|
| Total KE gained \(= \dfrac{1}{2} \times 6 \times 4.5^2\ (= 60.75)\ (\text{J})\) | B1 | 3.1b |
| Max friction \(\mu 2g\cos\theta\ (= \mu \times 2 \times 9.8 \times \cos\theta = 15.68\mu)\) | B1 | 3.1b |
| Work done against friction \(= 3 \times F_{\max}\ (= 47.04\mu)\) | B1ft | 3.4 |
| Work-energy equation: their GPE lost = their KE gained + their WD against friction | M1 | 3.4 |
| \(82.32 = 60.75 + 47.04\mu\) | A1 | 1.1b |
| \(\mu = 0.459\ (0.46)\) | A1 | 1.1b |
| (6) |
Notes
B1: Gain in KE for the system (not just for one block)
B1: Correct unsimplified expression for \(F_{\max}\) seen or implied
B1ft: Correct expression for work done: follow their \(F_{\max}\). This is dependent on them having found an expression for \(F_{\max}\)
M1: Complete method using work-energy to form an equation in \(\mu\). Require all terms (needs to consider the KE and GPE of both blocks). Dimensionally correct. Condone sign errors.
A1: Correct unsimplified equation in \(\mu\)
A1: 3 sf or 2 sf only
NB: It is possible to find the value of \(\mu\) by finding the tension in the string and forming a work-energy equation for particle \(B\), but in this case the first
B1 is for KE of \(B\) and correct tension (25.7(N))
B1 for \(F_{\max}\)
B1ft is for work done by the tension in the string and against friction
M1 for \(3 \times 25.7 = 20.25 + 35.28 + 3 \times 15.68\mu\) O.E.
| Scheme | Marks | AO |
|---|---|---|
| Work-energy equation for \(A\): | M1 | 3.4 |
| \(\dfrac{1}{2} \times 2 \times 4.5^2 = 2g\sin\theta \times d + 2g\cos\theta \times \mu d\) \(\left(= 19.6 \times \dfrac{3}{5} \times d + 19.6 \times \dfrac{4}{5} \times \mu d\right)\) | A1ft A1ft | 1.1b 1.1b |
| \(d = 1.07\ (1.1)\) | A1 | 1.1b |
| (4) | ||
| (13 marks) |
Notes
M1: All terms required. Dimensionally correct. Condone sign errors and sin / cos confusion.
If the equation uses \(d + 3\) in place of \(d\) in the PE term it is correct if it also includes a term for the initial PE.
If the equation uses \(d + 3\) in place of \(d\) in the term for work done then it scores M0.
A1 A1: Unsimplified equation in \(d\) and \(\mu\) with at most one error
Correct unsimplified equation in \(d\) and \(\mu\)
The ft is on their \(\mu\) if they have substituted a value.
A1: 3 sf or 2 sf only