Law of Motion III

Inclined Plane

i) Case - 1 
Body sliding freely on inclined plane.
Perpendicular to the plane
N = mgCosθ (since body is at rest)

Parallel to the plane mgCos θ
mgSinθ = ma θ mg

ii) Case - 2
Body pulled parallel to the inclined plane.
Perpendicular to the plane
N = mgCosθ (since body is at rest)

Parallel to the plane θ
F - mgSinθ = ma

iii) Case - 3
Body pulled parallel to the inclined plane but accelerating downwards.
Perpendicular to the plane F
N = mgCosθ (since body is at rest)

Parallel to the plane
mgSinθ - F = ma

iv) Case - 4
Body accelerating up the incline under the effect of two forces acting parallel to the incline.
Perpendicular to the plane
N = mgCosθ (since body is at rest)

Parallel to the plane mgSinθ θ

F₁ - F₂ - mgSinθ = ma

v) Case - 5
Body accelerating up the incline under the effect of horizontal force.
Perpendicular to the plane
N = mgCosθ + F₁Sinθ (since body is at rest)

Parallel to the plane
F₁Cosθ - mgSinθ = ma

vi) Case - 6
Body accelerating down the incline under the effect of horizontal force and gravity.
Perpendicular to the plane
N + FSinθ = mgCosθ (since body is at rest)

Parallel to the plane mgSinθ
FCosθ + mgSinθ = ma

vii) Case - 7
Body accelerating up the incline under the effect of two horizontal forces acting on opposite sides of a body and gravity.
Perpendicular to the plane F₁
N + F₁Sinθ = mgCosθ + F₂Sinθ(since body is at rest)

Parallel to the plane mgSinθ
F₂Cosθ - F₁Cosθ - mgSinθ = ma

Vertical Plane
i) Case - 1
Body pushed against the vertical plane by horizontal force and moving vertically downward.
For horizontal direction
mg = ma (since body is at rest)

For vertical direction
F = N

ii) Case - 2
Body pushed against the vertical plane by horizontal force and pulled vertically upward.
For vertical direction
F₂ - mg = ma
For horizontal direction (since body is at rest)

N = F₁

iii) Case - 3
Body pushed against the vertical plane by inclined force and accelerates vertically upward.
For horizontal direction θ
N = FSinθ (since body is at rest)

For vertical direction
FCosθ – mg = ma

iv) Case - 4
Body pushed against the vertical plane by inclined force and accelerates vertically downward.
For horizontal direction FSinθ
N = FSinθ (since body is at rest)

For vertical direction FCosθ
FCosθ + mg = ma

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