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 (mass 4 kg), we consider its equation of motion:
[ 4g - T = 4a ]
where ( g ) is the acceleration due to gravity and ( a ) is the acceleration of the particles.
Step 2
Equation of motion for A
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Answer
For particle A (mass 2 kg), the equation of motion is expressed as:
[ T - F - 2g \sin(30°) = 2a ]
Here, ( F = \mu R ) and ( R ) is the normal reaction force.
Step 3
Resolving perpendicular to the plane
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Answer
To resolve the forces perpendicular to the plane:
[ R = 2g \cos(30°) ]
where ( \mu = \frac{1}{\sqrt{3}} ) gives us the frictional force.
Step 4
Using \( F = \mu R \)
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Substituting the expression for ( R ):
[ F = \mu R = \frac{1}{\sqrt{3}} (2g \cos(30°)) ]
We calculate the value of frictional force and substitute back into the equation for A.
Step 5
Final Equations
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Now, substituting ( F ) back into the second equation:
Combining these equations gives us all the terms required to isolate ( T ) and find its value.
Step 6
Solving the equations
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Answer
Using the equations from the system:
From the equation of motion for B:
[ T = 4g - 4a ]
Substituting this into the equation for A leads to:
[ 2T - 4g - T + F = 2a ]
This will allow us to solve for the tension ( T ) after substituting the known values of ( g ) and simplifying.
