M1 June 2018 Q5

EdexcelOld spec10 marksConnected Particles Forces & Newton's Laws

Figure 3: lift hanging from a vertical cable with a woman standing on its floor — Figure 3

A lift of mass 250 kg is being raised by a vertical cable attached to the top of the lift. A woman of mass 60 kg stands on the horizontal floor inside the lift, as shown in Figure 3. The lift ascends vertically with constant acceleration 2 m s\(^{-2}\). There is a constant downwards resistance of magnitude 100 N on the lift. By modelling the woman as a particle,

(a) find the magnitude of the normal reaction exerted by the floor of the lift on the woman. (3)

The tension in the cable must not exceed 10 000 N for safety reasons, and the maximum upward acceleration of the lift is 3 m s\(^{-2}\). A typical occupant of the lift is modelled as a particle of mass 75 kg and the cable is modelled as a light inextensible string. There is still a constant downwards resistance of magnitude 100 N on the lift.

(b) Find the maximum number of typical occupants that can be safely carried in the lift when it is ascending with an acceleration of 3 m s\(^{-2}\). (7)

Mark scheme (a)

Scheme	Marks
\(R - 60g = 60 \times 2\)	M1A1
\(R = 708\) N or 710 N (must be positive)	A1
	(3)

Notes

M1 for equation in \(R\) only, with usual rules

First A1 for a correct equation

Second A1 for 710 or 708 (N not needed)

Mark scheme (b)

Scheme	Marks
\(75n\)	B1
\(10000 - Mg - 100 = M \times 3\)	M1A2
using \(M = 250 + 75n \Rightarrow n = 6.9..\)	DM1A1
so 6 people	A1ft
	(7)
	(10 marks)

Notes

B1 for \(75n\) oe seen or implied

First M1 for an equation in one unknown in the form \(10000 - Mg - 100 = M \times a\) with usual rules (must be using 10000) where \(M\) can be any (relevant) number e.g. 250, 75, etc

First A1 and second A1 for a correct equation with \(a = 3\), A1A0 if one error (e.g. Use of \(a = 2\) loses 1 A mark)

Second DM1, dependent on first M1, for using \(M = 250 + 75n\) and solving for \(n\)

Third A1 for 6.9… (A0 for 7)

Fourth A1ft for no. of people, ft on their \(n\) value (A0 for < 7)

N.B. If no incorrect work seen, the third A mark can be implied by a correct answer (\(n = 6\))

SC: They may use Trial and Error to find the critical value of \(n\), by writing down equations for the tension when \(n = 1, 2, 3, \ldots\) until the tension exceeds 10000 oe

This method can score the final DM1 A1 A1 if done fully correctly up to and including \(n = 7\), with a correct answer given.

It could also score some or all of the first 4 marks.