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Paper 4 (9164, Section B) · November 2009 · Friction and Inclined Planes

A block of mass 3.5 kg is released from rest at point A on a plane inclined at angle α\alpha to the horizontal, where tan⁡α=34\tan\alpha = \dfrac{3}{4}. It slides down to point B at the base of the plane. The coefficient of friction between the block and the plane is 14\dfrac{1}{4}. Take g=9.81g = 9.81 ms−2^{-2}.

Calculate the acceleration of the block down the plane, in ms−2^{-2}.

Model answer

3.9

Also accepted: 3,9, 3.92, 3,92, 3.924, 3,924

Explanation

From tan⁡α=34\tan\alpha=\dfrac{3}{4} the 3, 4, 5 triangle gives sin⁡α=35=0.6\sin\alpha=\dfrac{3}{5}=0.6 and cos⁡α=45=0.8\cos\alpha=\dfrac{4}{5}=0.8.

Perpendicular to the plane the block does not move, so R=mgcos⁡αR=mg\cos\alpha and the friction, which opposes the sliding and so acts up the plane, is μmgcos⁡α\mu mg\cos\alpha.

Along the plane, mgsin⁡α−μmgcos⁡α=mamg\sin\alpha-\mu mg\cos\alpha=ma. The mass cancels from every term:

a=g(sin⁡α−μcos⁡α)=9.81(0.6−14(0.8))=9.81(0.6−0.2)=9.81(0.4)=3.924a=g(\sin\alpha-\mu\cos\alpha)=9.81\left(0.6-\tfrac{1}{4}(0.8)\right)=9.81(0.6-0.2)=9.81(0.4)=3.924 ms−2^{-2},

which is 3.9 ms−2^{-2} correct to 2 significant figures. Notice the 3.5 kg never enters the answer.

Derived from ZIMSEC Mathematics 9164/4 Paper 4, November 2009, Q14

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