M4 June 2007 Q3

EdexcelOld spec12 marksWork-Energy Principle

Figure 1: framework of rods AB and BC, each 2a long, at right angles at B, with a light rod joining their mid-points; AB at angle theta to the downward vertical through A — Figure 1

A framework consists of two uniform rods \(AB\) and \(BC\), each of mass \(m\) and length \(2a\), joined at \(B\). The mid-points of the rods are joined by a light rod of length \(a\sqrt{2}\), so that angle \(ABC\) is a right angle. The framework is free to rotate in a vertical plane about a fixed smooth horizontal axis. This axis passes through the point \(A\) and is perpendicular to the plane of the framework. The angle between the rod \(AB\) and the downward vertical is denoted by \(\theta\), as shown in Fig. 1.

(a) Show that the potential energy of the framework is \[-mga(3\cos\theta + \sin\theta) + \text{constant}.\] (4)

(b) Find the value of \(\theta\) when the framework is in equilibrium, with \(B\) below the level of \(A\). (4)

Framework hanging from A: AB of length 2a, BC of length 2a at right angles at B, mid-points joined — Mark scheme (a)

Scheme	Marks
\(V = -mga\cos\theta - mg(2a\cos\theta + a\sin\theta)\)	M1A1A1
\(= -mga(3\cos\theta + \sin\theta)\quad (+const)\ *\)	A1
	(4)

Mark scheme (b)

Scheme	Marks
\(\dfrac{dV}{d\theta} = -mga(-3\sin\theta + \cos\theta)\)	M1A1
\(= 0\ \Rightarrow\ \tan\theta = \dfrac{1}{3}\)	M1
\(\Rightarrow \theta = 0.32(1)^c\) or \(18.4^o\) accept awrt	A1
	(4)

Notes

M1 Attempt to differentiate V. Condone errors in signs and in constants.

A1 Derivative correct

M1 Set derivative = 0 and rearrange to a single trig function in \(\theta\)

A1 Solve for \(\theta\)

or M1A1 find the position of the center of mass

M1A1 form and solve trig equation for \(\theta\)

Mark scheme (c)

Scheme	Marks
\(\dfrac{d^2V}{d\theta^2} = -mga(-3\cos\theta - \sin\theta)\)	M1A1
\(= mga(3\cos\theta + \sin\theta)\)
Hence, when \(\theta = 0.32^c\), \(\ \dfrac{d^2V}{d\theta^2} \gt 0\)	M1
i.e. stable	A1
	(4)
	(12 marks)

Notes

M1 Differentiate to obtain the second derivative

A1 Derivative correct

M1 Determine the sign of the second derivative

A1 Correct conclusion. cso

Or: M1 Find the value of \(\dfrac{dV}{d\theta}\) on both sides of the minimum point

A1 signs correct

M1 Use the results to determine the nature of the turning point

A1 Correct conclusion, cso.

These 4 marks are dependent on the use of derivatives