October 2020 Paper 1 Q15

OCR MEICurrent spec9 marksResolving Forces Vectors

15 Fig. 15 shows a particle of mass \(m\) kg on a smooth plane inclined at \(30^\circ\) to the horizontal. Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel and perpendicular to the plane, in the directions shown.

Fig. 15: particle on a plane inclined at 30 degrees; i points up the plane, j perpendicular to the plane away from it — Fig. 15

(a) Express the weight \(\mathbf{W}\) of the particle in terms of \(m\), \(g\), \(\mathbf{i}\) and \(\mathbf{j}\). [2]

The particle is held in equilibrium by a force \(\mathbf{F}\), and the normal reaction of the plane on the particle is denoted by \(\mathbf{R}\). The units for both \(\mathbf{F}\) and \(\mathbf{R}\) are newtons.

(b) Write down an equation relating \(\mathbf{W}\), \(\mathbf{R}\) and \(\mathbf{F}\). [1]

show that \(m = 1.22\) correct to 3 significant figures,
find the magnitude of \(\mathbf{R}\).

[6]

Mark scheme (a)

Scheme	Marks	AO
\(\mathbf{W} = (-mg\sin 30^\circ)\mathbf{i} + (-mg\cos 30^\circ)\mathbf{j}\)	M1	3.1b
\(\left[\mathbf{W} = \left(-\dfrac{1}{2}mg\right)\mathbf{i} + \left(-\dfrac{\sqrt{3}}{2}mg\right)\mathbf{j}\right]\)	A1	2.5
	[2]

Notes

M1: Attempting to resolve the weight. Allow sin/cos interchange and sign errors for the method mark.
\(mg\) must be seen for the method mark.

A1: All correct in this vector form

Mark scheme (b)

Scheme	Marks	AO
\(\mathbf{W} + \mathbf{R} + \mathbf{F} = \mathbf{0}\)	B1	2.5
	[1]

Notes

B1: Allow any rearrangement of this.
Allow if their expression for \(\mathbf{W}\) is used instead of \(\mathbf{W}\)

Mark scheme (c)

Scheme	Marks	AO
\(\mathbf{R} = R\mathbf{j}\) \(\left[(-mg\sin 30^\circ)\mathbf{i} + (-mg\cos 30^\circ)\mathbf{j} + (6\mathbf{i} + 8\mathbf{j}) + R\mathbf{j} = \mathbf{0}\right]\)	B1	3.1b
i component \(-mg\sin 30^\circ + 6 = 0\)	M1	3.1a
giving \(m = 1.22\) to 3 sf	A1 A1	1.1 1.1
j component: \(-12\cos 30 + 8 + R = 0\)	M1	3.1a
\(R = 2.39\) so magnitude is 2.39 N	A1	3.2a
	[6]

Notes

B1: Allow for any clear indication that \(\mathbf{R}\) is a multiple of \(\mathbf{j}\) or that it has no component in the \(\mathbf{i}\) direction
May be implied with an equation for the i direction with two terms and an equation in the j direction with three terms

M1: Forming equation from their \(\mathbf{i}\) terms, or equivalent by resolving parallel to the plane. FT their \(\mathbf{W}\)

A1: Correct equation in \(\mathbf{i}\) direction

A1: AG

M1: Equation from the \(\mathbf{j}\) terms (must include all three terms), oe, and using value of \(m\)
12 is the value for \(mg\). \(1.22 \times 9.8 = 11.956\)

A1: Accept arwt 2.4