A2 June 2023 Q6
6.
A hollow right circular cone, of internal base radius 0.6 m and height 0.8 m, is fixed with its axis vertical and its vertex \(V\) pointing downwards, as shown in Figure 4.
A particle \(P\) of mass \(m\) kg moves in a horizontal circle of radius 0.5 m on the rough inner surface of the cone.
The particle \(P\) moves with constant angular speed \(\omega\ \text{rad s}^{-1}\)
The coefficient of friction between the particle \(P\) and the inner surface of the cone is 0.25
Find the greatest possible value of \(\omega\) (9)
| Scheme | Marks | AO |
|---|---|---|
 | ||
| Resolve vertically | M1 | 3.4 |
| \(F\cos\theta + mg = R\sin\theta\) \((4F + 5mg = 3R)\) | A1 A1 | 1.1b 1.1b |
| Equation for motion towards centre | M1 | 3.4 |
| \(F\sin\theta + R\cos\theta = m \times 0.5\omega^2\) \(\left(3F + 4R = m \times 2.5\omega^2\right)\) | A1 A1 | 1.1b 1.1b |
| At max \(\omega\) \(R + 5mg = 3R \quad \left(R = \dfrac{5mg}{2}\right)\) \(\dfrac{3}{4}R + 4R = m \times 2.5\omega^2 \quad \left(19R = 10m\omega^2\right)\) | M1 | 1.2 |
| Solve for \(\omega\): \(19 \times \dfrac{5mg}{2} = 10m\omega^2\) | DM1 | 3.1a |
| \(\omega^2 = \dfrac{19g}{4} \Rightarrow \max\omega = 6.8\ \ (6.82)\) | A1 | 2.2a |
| (9) | ||
| (9 marks) |
Notes
M1: Need all terms. Condone sign errors and sin/cos confusion.
A1: Unsimplified equation with at most one error.
A1: Correct unsimplified equation.
M1: Need all terms. Condone sign errors and sin/cos confusion in \(R\) or their \(R\).
A1: Unsimplified equation with at most one error.
A1: Correct unsimplified equation in \(R\) or their \(R\).
NB: if \(F\) in wrong direction count this as one error (not one in each equation) and deduct one accuracy mark in the first equation affected.
Either / both equation(s) could be replaced with equations for resolving parallel and perpendicular to the surface. Perpendicular:
\(R = mg\sin\theta + mr\omega^2\cos\theta \quad \left(R = \dfrac{3}{5}mg + \dfrac{2}{5}m\omega^2\right)\)
Parallel: \(F + mg\cos\theta = mr\omega^2\sin\theta \quad \left(F + \dfrac{4}{5}mg = \dfrac{3}{10}m\omega^2\right)\)
If they have more than 2 equations, mark the correct equations. If they go on to use an incorrect equation then DM0.
M1: Use of \(F = \mu R\) to eliminate \(F\) or \(R\)
Condone inequality
DM1: Complete method including substitution of trig values to obtain a value for \(\omega\)
A1: 2 s.f or 3 s.f only \(\sqrt{\dfrac{19g}{4}}\) is A0
Must be an equation