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A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude P newtons - Edexcel - A-Level Maths Mechanics - Question 7 - 2010 - Paper 1
Question 7
A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude P newtons. The force acts in the vertical plane containing the li... show full transcript
Worked Solution & Example Answer:A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude P newtons - Edexcel - A-Level Maths Mechanics - Question 7 - 2010 - Paper 1
Step 1
(a) the magnitude of the normal reaction between the particle and the plane
Answer
To find the normal reaction R, we analyze the forces acting on the particle on the inclined plane.
Using the equilibrium of forces in the direction perpendicular to the incline:
R = rac{1}{3}F
Considering the forces:
From this, we can derive:
R ext{ cos }α - rac{R}{3} ext{ sin }α = 0.4g
Substituting an(α) = rac{3}{4} gives:
R ext{ cos }α - rac{1}{3}R rac{3}{4} = 0.4g
This simplifies to:
R ext{ cos }α - rac{1}{4} R = 0.4g
Solving for R yields:
R = rac{0.4g}{ ext{cos }α - rac{1}{4}}
Substituting the known values gives:
- Calculate : R = rac{0.4 imes 9.81}{ ext{cos } ext{arctan}(3/4) - 0.25}
Thus, the magnitude of the normal reaction between the particle and the plane is approximately 6.53 N or 6.5 N.
Step 2
(b) the value of P
Answer
To find the value of P, we consider the forces acting parallel to the incline:
Using the equilibrium of forces parallel to the incline:
From the previous part, we substitute R:
Replacing in our equation gives:
P - rac{R}{3} ext{ cos }α - R ext{ sin }α = 0
Substituting our known values for R, and solving for P gives: P = R ext{ sin }α + rac{R}{3} ext{ cos }α
Performing these calculations, we find:
- Substitute for the calculations: P = rac{2}{6}g = 5.66 ext{ N or } 5.7 ext{ N}
Thus, the value of P is approximately 5.66 N or 5.7 N.