What does 'smooth pulley' tell you?

The tension is the same on both sides; there is no friction at the pulley.

How do you find the force on a smooth pulley holding two vertical strings?

Add the tensions: 2T downwards.

What acceleration has a particle moving freely upwards after the string goes slack?

g downwards (-9.8 m s⁻² up positive).

Give the equation of motion for sliding down a rough slope.

mg\sin\theta-\mu mg\cos\theta=ma

Why do both connected particles have the same acceleration?

The string is inextensible and taut, so both move the same distance.

What do you do when a particle hits the ground and stops?

Start a new stage with new forces; the other particle's initial speed is the final speed of stage 1.

Connected particles, pulleys and inclined planesEdexcel International A Level Maths: Flashcards

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What does 'light inextensible string' tell you?

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What does 'light inextensible string' tell you?: Negligible mass so the tension is the same throughout, and both ends have the same acceleration.
What does 'smooth pulley' tell you?: The tension is the same on both sides; there is no friction at the pulley.
Write the formula for acceleration of two hanging masses $m_1>m_2$ .: $a=\frac{(m_1-m_2)g}{m_1+m_2}$
Write the formula for tension for two hanging masses $m_1$ , $m_2$ .: $T=\frac{2m_1m_2g}{m_1+m_2}$
How do you find the force on a smooth pulley holding two vertical strings?: Add the tensions: $2T$ downwards.
What is the tension in a string after a particle at its end lands and the string goes slack?: Zero.
What acceleration has a particle moving freely upwards after the string goes slack?: $g$ downwards ( $-9.8$ m s⁻² up positive).
What is the acceleration of a particle sliding on a smooth plane inclined at $\theta$ ?: $g\sin\theta$ down the slope.
What is the normal reaction on a particle on a plane inclined at $\theta$ with no other perpendicular force?: $R=mg\cos\theta$
State the friction law for a sliding particle.: $F=\mu R$ , opposing the motion.
Which way does friction act on a particle sliding up a slope?: Down the slope.
Give the equation of motion for sliding down a rough slope.: $mg\sin\theta-\mu mg\cos\theta=ma$
Why do both connected particles have the same acceleration?: The string is inextensible and taut, so both move the same distance.
What do you do when a particle hits the ground and stops?: Start a new stage with new forces; the other particle's initial speed is the final speed of stage 1.

Exam questions on Connected particles, pulleys and inclined planes

Two particles $A$ and $B$ , of masses $3$ kg and $5$ kg, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley and the particles hang vertically with the string taut. The system is released from rest. Take $g=9.8$ m s⁻².
Find the magnitude of the force exerted on the pulley by the string.2 marks
Particles $P$ and $Q$ , of masses $3$ kg and $2$ kg, are attached to the ends of a light inextensible string that passes over a smooth fixed pulley. Initially $Q$ is on horizontal ground and $P$ is $1.2$ m above the ground, with the string taut and vertical. The system is released from rest. After $P$ hits the ground it does not rebound. In the motion described, $Q$ does not reach the pulley. Take $g=9.8$ m s⁻².
After $P$ hits the ground, find the further distance that $Q$ rises before it first comes to rest.2 marks
A particle of mass $4$ kg is on a smooth plane inclined at $30^\circ$ to the horizontal. Take $g=9.8$ m s⁻².
The particle is released from rest. Find its acceleration down the plane.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).