M2 June 2007 Q4

EdexcelOld spec7 marksWork-Energy Principle

4.

Figure 2: particle A on a rough plane inclined at alpha, string over pulley P at the top, particle B hanging
Figure 2

Two particles \(A\) and \(B\), of mass \(m\) and \(2m\) respectively, are attached to the ends of a light inextensible string. The particle \(A\) lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan\alpha = \dfrac{3}{4}\). The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The particle \(B\) hangs freely below \(P\), as shown in Figure 2. The particles are released from rest with the string taut and the section of the string from \(A\) to \(P\) parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\dfrac{5}{8}\). When each particle has moved a distance \(h\), \(B\) has not reached the ground and \(A\) has not reached \(P\).

(a) Find an expression for the potential energy lost by the system when each particle has moved a distance \(h\). (2)

When each particle has moved a distance \(h\), they are moving with speed \(v\). Using the work-energy principle,

(b) find an expression for \(v^2\), giving your answer in the form \(kgh\), where \(k\) is a number. (5)
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