AS June 2024 Q2

EdexcelCurrent spec10 marksHorizontal Circular Motion

Figure 2: hollow hemisphere with centre O and radius a, axis vertical; ball B moves in a horizontal circle of centre C and radius r on the inner surface, with C a distance h vertically below O — Figure 2

A thin hollow hemisphere, with centre \(O\) and radius \(a\), is fixed with its axis vertical, as shown in Figure 2.

A small ball \(B\) of mass \(m\) moves in a horizontal circle on the inner surface of the hemisphere. The circle has centre \(C\) and radius \(r\). The point \(C\) is vertically below \(O\) such that \(OC = h\).

The ball moves with constant angular speed \(\omega\)

The inner surface of the hemisphere is modelled as being smooth and \(B\) is modelled as a particle. Air resistance is modelled as being negligible.

(a) Show that \(\omega^2 = \dfrac{g}{h}\) (6)

Given that the magnitude of the normal reaction between \(B\) and the surface of the hemisphere is \(3mg\)

(b) find \(\omega\) in terms of \(g\) and \(a\). (3)

force diagram: ball on the inner surface with mg vertically down and R along the radius towards the centre, at angle theta above the horizontal radius r; depth h and radius a marked — Mark scheme (a)

Scheme	Marks	AO

Resolve horizontally	M1	3.1b
\(mr\omega^2 = R\cos\theta\)	A1	1.1b
Resolve vertically	M1	3.1b
\(mg = R\sin\theta\)	A1	1.1b
\(\dfrac{g}{r\omega^2} = \tan\theta = \dfrac{h}{r}\)	DM1	1.1b
\(\Rightarrow \omega^2 = \dfrac{g}{h}\) *	A1*	2.2a
	(6)

Mark scheme (b)

Scheme	Marks	AO
Resolve horizontally or vertically	M1	2.1
\(ma\cos\theta\,\omega^2 = 3mg\cos\theta \Rightarrow a\omega^2 = 3g\) or \(mg = 3mg\sin\theta \Rightarrow g = 3g \times \dfrac{g}{a\omega^2}\)	A1	1.1b
\(\omega = \sqrt{\dfrac{3g}{a}}\)	A1	1.1b
	(3)

Notes

M1: Resolve and substitute for \(R\) and \(r\). Dimensionally correct. Condone sine/cosine confusion

A1: Correct equation in \(a\), \(g\) and \(\omega\)

A1: Correct only

Mark scheme (c)

Scheme	Marks	AO
Have ignored the dimensions of \(B\)	B1	3.5b
	(1)
	(10 marks)

Notes

B1: Or equivalent or acceptable alternatives e.g. have ignored spin of \(B\)
B0 for weight acts through a point
B0 if any incorrect extras