A2 June 2023 Q4
4.
A smooth hemisphere of radius \(a\) is fixed on a horizontal surface with its plane face in contact with the surface. The centre of the plane face of the hemisphere is \(O\).
A particle \(P\) of mass \(M\) is disturbed from rest at the highest point of the hemisphere.
When \(P\) is still on the surface of the hemisphere and the radius from \(O\) to \(P\) is at an angle \(\theta\) to the vertical,
- the speed of \(P\) is \(v\)
- the normal reaction between the hemisphere and the particle is \(R\), as shown in Figure 2.
| Scheme | Marks | AO |
|---|---|---|
| Conservation of energy: | M1 | 3.1a |
| \(\dfrac{1}{2}Mv^2 = Mga(1 - \cos\theta)\) | A1 | 1.1b |
| Equation of motion: | M1 | 3.1a |
| \(\dfrac{Mv^2}{a} = Mg\cos\theta - R\) | A1 | 1.1b |
| Solve for \(R\): | DM1 | 2.1 |
| \(R = Mg\cos\theta - \dfrac{Mv^2}{a} = Mg\cos\theta - 2Mg(1 - \cos\theta)\) \(= 3Mg\cos\theta - 2Mg = Mg(3\cos\theta - 2)\) * | A1* | 2.2a |
| (6) |
Notes
M1: Dimensionally correct equation. All relevant terms. Condone sign error(s) and sin/cos confusion.
A1: Correct unsimplified equation.
M1: Dimensionally correct equation. All relevant terms. Condone sign error(s) and sin/cos confusion.
A1: Correct unsimplified equation.
If they have more than 2 equations, mark the correct equations. If they go on to use an incorrect equation then DM0.
DM1: Complete method to obtain expression for \(R\)
A1*: Obtain given answer from full and correct working.
| Scheme | Marks | AO |
|---|---|---|
| \(R = 0 \Rightarrow \cos\theta = \dfrac{2}{3}\) | B1 | 1.1b |
| \(v^2 = 2ga(1 - \cos\theta)\) | M1 | 3.1a |
| \(v^2 = \dfrac{2}{3}ga, \quad v = \sqrt{\dfrac{2ga}{3}}\) | A1 | 1.1b |
| (3) | ||
| (9 marks) |
Notes
B1: Seen or implied.
M1: Complete method to obtain \(v\) or \(v^2\)
A1: Any equivalent form.
Allow \(0.82\sqrt{ga}\) or better.