As shown in the diagram, particles $$\(A\)$$ and $$\(B\)$$ of masses $$\(2 \mathrm{~kg}\)$$ and $$\(3 \mathrm{~kg}\)$$ respectively are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the top of two inclined planes. Particle $$\(A\)$$ is on plane $$\(P\)$$, which is inclined at an angle of $$\(10^{\circ}\)$$ to the horizontal. Particle $$\(B\)$$ is on plane $$\(Q\)$$, which is inclined at an angle of $$\(20^{\circ}\)$$ to the horizontal. The string is taut, and the two parts of the string are parallel to lines of greatest slope of their respective planes. It is given instead that both planes are smooth and that the particles are released from rest at the same horizontal level. Find the time taken until the difference in the vertical height of the particles is $$\(1 \mathrm{~m}\)$$. [You should assume that this occurs before $$\(A\)$$ reaches the pulley or $$\(B\)$$ reaches the bottom of plane $$\(Q\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_w20_qp_43 Year：2020 Question No：7(b)
Answer:
\(3 g \sin 20-T=3 a\) or \(T-2 g \sin 10=2 a\)
or System: \(3 g \sin 20-2 g \sin 10=5 a\)
\(a=\frac{(3 g \sin 20-2 g \sin 10)}{5}\)
\(a=1.3575 \ldots\)
\(h_{1}=x \sin 20\)
\(h_{2}=x \sin 10\)
\(x \sin 20+x \sin 10=1\)
\(\frac{1}{\sin 10+\sin 20}=0+\frac{1}{2} \times 1.3575 \times t^{2}\)
\(t=1.69\)
or System: \(3 g \sin 20-2 g \sin 10=5 a\)
\(a=\frac{(3 g \sin 20-2 g \sin 10)}{5}\)
\(a=1.3575 \ldots\)
\(h_{1}=x \sin 20\)
\(h_{2}=x \sin 10\)
\(x \sin 20+x \sin 10=1\)
\(\frac{1}{\sin 10+\sin 20}=0+\frac{1}{2} \times 1.3575 \times t^{2}\)
\(t=1.69\)
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.
4.4.2 use the relationship between mass and weight W = mg. In this component, questions are mainly numerical, and use of the approximate numerical value 10 (m ) for g is expected.
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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