June 2019 Paper 3 Mechanics Q4
4.
A ramp, \(AB\), of length 8 m and mass 20 kg, rests in equilibrium with the end \(A\) on rough horizontal ground.
The ramp rests on a smooth solid cylindrical drum which is partly under the ground. The drum is fixed with its axis at the same horizontal level as \(A\).
The point of contact between the ramp and the drum is \(C\), where \(AC = 5\) m, as shown in Figure 2.
The ramp is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle \(\theta\) to the horizontal, where \(\tan\theta = \dfrac{7}{24}\)
The ramp is modelled as a uniform rod.
The ramp is still in equilibrium in the position shown in Figure 2 but the ramp is not now modelled as being uniform.
Given that the centre of mass of the ramp is assumed to be closer to \(A\) than to \(B\),
| Scheme | Marks | AO |
|---|---|---|
| Drum smooth, or no friction, (therefore reaction is perpendicular to the ramp) | B1 | 2.4 |
| (1) |
Notes
B1: Ignore any extra incorrect comments.
| Scheme | Marks | AO |
|---|---|---|
| N.B. In (b), for a moments equation, if there is an extra \(\sin\theta\) or \(\cos\theta\) on a length, give M0 for the equation e.g. M(\(A\)): \(20g \times 4\cos\theta = 5N\sin\theta\) would be given M0A0 | ||
 | ||
| Possible equns \((\nearrow)\): \(F\cos\theta + R\sin\theta = 20g\sin\theta\) \((\nwarrow)\): \(N + R\cos\theta = 20g\cos\theta + F\sin\theta\) \((\uparrow)\): \(R + N\cos\theta = 20g\) \((\rightarrow)\): \(F = N\sin\theta\) M(\(A\)): \(20g \times 4\cos\theta = 5N\) M(\(B\)): \(3N + R \times 8\cos\theta = F \times 8\sin\theta + 20g \times 4\cos\theta\) M(\(C\)): \(R \times 5\cos\theta = F \times 5\sin\theta + 20g \times \cos\theta\) M(\(G\)): \(R \times 4\cos\theta = F \times 4\sin\theta + N\) | M1 A1 M1 A1 M1 A1 | 3.3 1.1b 3.4 1.1b 3.4 1.1b |
| (The values of the 3 unknowns are: \(N = 150.528\); \(F = 42.14784\); \(R = 51.49312\)) | ||
| Solve their 3 equations for \(F\) and \(R\) OR \(X\) and \(Y\) OR \(H\) and \(S\) | M1 | 1.1b |
| \(|\text{Force}| = \sqrt{R^2 + F^2}\) Main scheme OR \(= \sqrt{X^2 + Y^2}\) Alternative 1 OR \(= \sqrt{(H^2 + S^2 - 2HS\cos(90^\circ - \theta))}\) Alternative 2 | M1 | 3.1b |
| Magnitude = 67 or 66.5 (N) | A1 | 2.2a |
| (9) |
Notes
Generally 3 independent equations required so at least one moments equation.: M1A1M1A1M1A1.
More than 3 equations, give marks for the best 3. For each:
M1 All terms required. Must be dimensionally correct so if a length is missing from a moments equation it’s M0 Condone sin/cos confusion.
A1 For a correct equation (trig ratios do not need to be substituted and allow e.g. cos(24/25) if they recover
Enter marks on ePEN in order in which equations appear.
N.B. If reaction at \(C\) is not perpendicular to the ramp, can only score marks for M(\(C\))
Allow use of \((\mu R)\) for \(F\)
M1: All terms required. Must be dimensionally correct. Condone sin/cos confusion.
A1: Correct unsimplified equation
M1: All terms required. Must be dimensionally correct. Condone sin/cos confusion.
A1: Correct unsimplified equation
M1: All terms required, dim correct, condone sin/cos confusion
A1: Correct unsimplified equation
N.B. They can find \(F\) and \(R\) using only TWO equations, the 1st and 7th in the list. Mark the better equation as M2A2 (-1 each error). Mark the second equation as M1A1
Alt 1
M1: All terms required. Must be dimensionally correct. Condone sin/cos confusion.
A1: Correct unsimplified equation
M1: All terms required. Must be dimensionally correct. Condone sin/cos confusion.
A1: Correct unsimplified equation
M1: All terms required. Must be dimensionally correct. Condone sin/cos confusion.
A1: Correct unsimplified equation
N.B. They can find \(X\) and \(Y\) using only TWO equations, the 1st and 7th in the list. Mark the better equation as M2A2 (-1 each error). Mark the second equation as M1A1
Alt 2
M1: All terms required. Must be dimensionally correct. Condone sin/cos confusion.
A1: Correct unsimplified equation
M1: All terms required. Must be dimensionally correct. Condone sin/cos confusion.
A1: Correct unsimplified equation
M1: All terms required. Must be dimensionally correct.
A1: Correct unsimplified equation
N.B. They can find \(H\) and \(S\) using only TWO equations, the 1st and 7th in the list. Mark the better equation as M2A2 (-1 each error). Mark the second equation as M1A1
M1: Substitute for trig and solve for their two cpts.
This is an independent mark but must use 3 equations (unless it’s the special case when 2 is sufficient)
M1: Use Pythagoras to find magnitude (this is an independent M mark but must have found a value for \(F\) (or \(X\)) and a value for \(R\) (or \(Y\)))
OR a complete method to find magnitude e.g. cosine rule but must have found a value for \(H\) and a value for \(S\)
A1: Correct answer only
Alternative
Alternative 1: using cpts along ramp (\(X\)) and perp to ramp (\(Y\))
| Scheme | Marks | AO |
|---|---|---|
| Possible equations: \((\nearrow)\): \(X = 20g\sin\theta\) \((\nwarrow)\): \(Y + N = 20g\cos\theta\) \((\uparrow)\): \(X\sin\theta + Y\cos\theta + N\cos\theta = 20g\) \((\rightarrow)\): \(X\cos\theta = Y\sin\theta + N\sin\theta\) M(\(A\)): \(20g \times 4\cos\theta = 5N\) M(\(B\)): \(20g \times 4\cos\theta = 8Y + 3N\) M(\(C\)): \(20g \times \cos\theta = 5Y\) M(\(G\)): \(4Y = N \times 1\) | M1 A1 M1 A1 M1 A1 | 3.3 1.1b 3.4 1.1b 3.4 1.1b |
| (The values of the 3 unknowns are: \(N = 150.528\); \(X = 54.88\); \(Y = 37.632\)) |
Alternative
Alternative 2: using horizontal cpt (\(H\)) and cpt perp to ramp (\(S\))
| Scheme | Marks | AO |
|---|---|---|
| \((\nearrow)\): \(H\cos\theta = 20g\sin\theta\) \((\nwarrow)\): \(S + N = H\sin\theta + 20g\cos\theta\) \((\uparrow)\): \(S\cos\theta + N\cos\theta = 20g\) \((\rightarrow)\): \(H = S\sin\theta + N\sin\theta\) M(\(A\)): \(20g \times 4\cos\theta = 5N\) M(\(B\)): \(20g \times 4\cos\theta + H \times 8\sin\theta = 8S + 3N\) M(\(C\)): \(20g \times \cos\theta + H \times 5\sin\theta = 5S\) M(\(G\)): \(4S = N \times 1 + H \times 4\sin\theta\) | M1 A1 M1 A1 M1 A1 | 3.3 1.1b 3.4 1.1b 3.4 1.1b |
| (The values of the 3 unknowns are: \(N = 150.528\); \(H = 57.1666…\); \(S = 53.638666…\)) |
| Scheme | Marks | AO |
|---|---|---|
| Magnitude of the normal reaction (at \(C\)) will decrease. | B1 | 3.5a |
| (1) | ||
| (11 marks) |
Notes
B1: Ignore reasons