Two particles $$\(A\)$$ and $$\(B\)$$ of masses $$\(2 \mathrm{~kg}\)$$ and $$\(3 \mathrm{~kg}\)$$ respectively are connected by a light inextensible string. Particle $$\(B\)$$ is on a smooth fixed plane which is at an angle of $$\(18^{\circ}\)$$ to horizontal ground. The string passes over a fixed smooth pulley at the top of the plane. Particle $$\(A\)$$ hangs vertically below the pulley and is $$\(0.45 \mathrm{~m}\)$$ above the ground (see diagram). The system is released from rest with the string taut. When $$\(A\)$$ reaches the ground, the string breaks. Find the total distance travelled by $$\(B\)$$ before coming to instantaneous rest. You may assume that $$\(B\)$$ does not reach the pulley. ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................
Exam No：9709_w21_qp_41 Year：2021 Question No：7
Answer:
Particle \(A: 2 g-T=2 a\)
Particle \(B: T-3 g \sin 18=T-9.27=3 a\)
System: \(2 g-3 g \sin 18=2 g-9.27=(2+3) a\)
\(a=2.145898034\)
\({[5 a=10.72949017] }\)
\(v^{2}=2 \times a \times 0.45\)
\(v^{2}=2 \times 2.145898034 \times 0.45=1.931308 \cdots\)
\({[v=1.389715162] }\)
\(T=0, \pm 3 g \sin 18=3 a\)
\({[a=\pm 3.0901699] }\)
\([0=1.93-2 \times 3.09 \times s] \quad[s=0.312]\)
Total distance moved by \(B=0.45+0.312=0.762 \mathrm{~m}\)
Alternative method for question
Attempt PE loss as \(A\) reaches the ground
PE loss \(=2 g \times 0.45-3 g \times 0.45 \sin 18\)
\([=4.82827]\)
\(2 g \times 0.45-3 g \times 0.45 \sin 18=\frac{1}{2} \times(2+3) v^{2}\)
Solve for \(v^{2}\)
\(v^{2}=1.931308 \ldots \quad[v=1.389715162]\)
PE gain \(=3 g \times s \sin 18\)
\(3 g \times s \sin 18=\frac{1}{2} \times 3 \times 1.931308 \quad[s=0.312]\)
Total distance moved by \(B=0.45+0.312=0.762 \mathrm{~m}\)
Particle \(B: T-3 g \sin 18=T-9.27=3 a\)
System: \(2 g-3 g \sin 18=2 g-9.27=(2+3) a\)
\(a=2.145898034\)
\({[5 a=10.72949017] }\)
\(v^{2}=2 \times a \times 0.45\)
\(v^{2}=2 \times 2.145898034 \times 0.45=1.931308 \cdots\)
\({[v=1.389715162] }\)
\(T=0, \pm 3 g \sin 18=3 a\)
\({[a=\pm 3.0901699] }\)
\([0=1.93-2 \times 3.09 \times s] \quad[s=0.312]\)
Total distance moved by \(B=0.45+0.312=0.762 \mathrm{~m}\)
Alternative method for question
Attempt PE loss as \(A\) reaches the ground
PE loss \(=2 g \times 0.45-3 g \times 0.45 \sin 18\)
\([=4.82827]\)
\(2 g \times 0.45-3 g \times 0.45 \sin 18=\frac{1}{2} \times(2+3) v^{2}\)
Solve for \(v^{2}\)
\(v^{2}=1.931308 \ldots \quad[v=1.389715162]\)
PE gain \(=3 g \times s \sin 18\)
\(3 g \times s \sin 18=\frac{1}{2} \times 3 \times 1.931308 \quad[s=0.312]\)
Total distance moved by \(B=0.45+0.312=0.762 \mathrm{~m}\)
Knowledge points:
4.2.4 use appropriate formulae for motion with constant acceleration in a straight line. （Questions may involve setting up more than one equation, using information about the motion of different particles.）
4.4.1 apply Newton’s laws of motion to the linear motion of a particle of constant mass moving under the action of constant forces, which may include friction, tension in an inextensible string and thrust in a connecting rod If any other forces resisting motion are to be considered (e.g. air resistance) this will be indicated in the question.
4.4.3 solve simple problems which may be modelled as the motion of a particle moving vertically or on an inclined plane with constant acceleration Including, for example, motion of a particle on a rough plane where the acceleration while moving up the plane is different from the acceleration while moving down the plane.
4.4.4 solve simple problems which may be modelled as the motion of connected particles. e.g. particles connected by a light inextensible string passing over a smooth pulley, or a car towing a trailer by means of either a light rope or a light rigid tow- bar.
Solution:
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