A fixed rough plane is inclined at 30° to the horizontal. A small smooth pulley P is fixed at the top of the plane. Two particles A and B, of mass 2 kg and 4 kg respectively, are attached to the ends of a light inextensible string which passes over the pulley P. The part of the string from A to P is parallel to a line of greatest slope of the plane and B hangs freely below P, as shown in Figure 2. The coefficient of friction between A and the plane is \( \mu = \frac{1}{\sqrt{3}} \). Initially A is held at rest on the plane. The particles are released from rest with the string taut and A moves up the plane. Find the tension in the string immediately after the particles are released.

A-Level Edexcel Maths Mechanics Moments: A fixed rough plane is inclined

A fixed rough plane is inclined at 30° to the horizontal - Edexcel - A-Level Maths Mechanics - Question 3 - 2013 - Paper 1

Question 3

A fixed rough plane is inclined at 30° to the horizontal. A small smooth pulley P is fixed at the top of the plane. Two particles A and B, of mass 2 kg and 4 kg resp... show full transcript

Worked Solution & Example Answer:A fixed rough plane is inclined at 30° to the horizontal - Edexcel - A-Level Maths Mechanics - Question 3 - 2013 - Paper 1

Step 1

Equation of motion for B:

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Answer

For particle B (4 kg), applying Newton's second law:

[ 4g - T = 4a ]

Where:

( T ) is the tension in the string,
( g ) is the acceleration due to gravity (approximately 9.81 m/s²),
( a ) is the acceleration of the system.

Step 2

Equation of motion for A:

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For particle A (2 kg), reflecting upon the forces along the inclined plane:

[ T - F - 2g\sin(30°) = 2a ]

Where ( F = \mu R ) is the frictional force acting against the motion, and ( R = 2g\cos(30°) ).

Step 3

Resolution perpendicular to the plane:

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To resolve the normal force for A:

[ R = 2g\cos(30°) = 2 \times 9.81 \times \frac{\sqrt{3}}{2} = 9.81 \sqrt{3} ]

Step 4

Finding frictional force F:

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Using the known friction coefficient:

[ F = \mu R = \frac{1}{\sqrt{3}} \times 9.81 \sqrt{3} = 9.81 ]

Step 5

Substituting values:

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Now substituting ( R ) and ( F ) into the equation for particle A:

[ T - 9.81 - 2g \sin(30°) = 2a ] This simplifies to find the unknown tension ( T ).

A fixed rough plane is inclined at 30° to the horizontal - Edexcel - A-Level Maths Mechanics - Question 3 - 2013 - Paper 1

Worked Solution & Example Answer:A fixed rough plane is inclined at 30° to the horizontal - Edexcel - A-Level Maths Mechanics - Question 3 - 2013 - Paper 1

Equation of motion for B:

Equation of motion for A:

Resolution perpendicular to the plane:

Finding frictional force F:

Substituting values:

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