The diagram shows a track ABCD which lies in a vertical plane. The section AB is a straight line inclined at an angle of 30° to the horizontal and is smooth. The section BC is a horizontal straight line and is rough. The section CD is a straight line inclined at an angle of 30° to the horizontal and is rough. The lengths AB, BC and CD are each 2m. A particle is released from rest at A. The coefficient of friction between the particle and both BC and CD is μ. There is no change in the speed of the particle when it passes through either of the points B or C.

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [B1] For CD R = mg cos 30. (May be seen in later working without.)
2. [M1] Use of F = 0.1R for either BC or CD. FBC = 0.1mg OR FCD = 0.1mg cos 30.
3. [A1] For sum of work done by friction and the change in PE. Note: Allow terms on different sides of a work energy equation as long as they have different signs.
4. [M1] Attempt at work energy equation with five relevant terms (dimensionally correct). Allow sign errors. Allow sin/cos errors but must be consistent. Note: Initial PE = mg.
5. [A1] d = 1.28 m or 15(10-√3)/97 [1.27854...]
6. [M1] Attempt at work energy equation with four relevant terms (dimensionally correct). Allow sign errors. Allow sin/cos errors but must be consistent.
7. [A1] μ = 0.174 or (4-√3)/13 [0.174457...]
8. [M1] For difference between the change in PE and the work done by friction. Note: Allow terms on different sides of a work energy equation as long as both have the same sign. Allow sin/cos errors but must be consistent. Using μ, their μ or the correct value of μ to at least 2 sf. Must be as part of an attempt to find speed, not μ, although this could be the first step.
9. [A1] Speed = 2.64ms⁻¹ [2.64164...]