Grade 10: Forces and Equilibrium

• Study notes for Grade 10 covering forces and equilibrium

Concepts

• Resultant of forces
• Resolution of a force
• Lami's rule
• Triangle of forces
• Equilibrium of co-planar forces acting at a point

Force

• Defined as the effect of a natural body upon another through pushing, attraction, pressure, or repulsion.
• A natural body consists of material (mass) and has a volume not equal to zero.
• Forces are vector quantities.
• The direction of force aligns with the direction of acceleration it causes.
• The net force on a body equals the vector sum of all forces acting upon it.

Resultant of Forces

• The resultant force is a single force that achieves the same effect as two or more forces combined.

Finding the Resultant of Two Forces Meeting at a Point (Graphically)

• If two forces, F1 and F2, meet at a point and are represented in magnitude and direction by two sides of a parallelogram meeting at this point, their resultant R is represented in magnitude and direction by the diagonal of the parallelogram starting from the same point.
• R = F1 + F2

Finding the Resultant of Two Forces Meeting at a Point (Analytically)

• If θ is the angle between the resultant R and force F₁, the magnitude and direction are given by:
• R = √(F1)² + (F2)² + 2F1F2 cos α, where α is the angle between F1 and F2.
• tan θ = (F2 sin α) / (F1 + F2 cos α)

Special Cases

• If two forces are perpendicular (α = 90°):
  • R = √(F1)² + (F2)², tan θ = F2 / F1
• If two forces are equal in magnitude (F1 = F2 = F):
  • R = 2F cos(α/2), θ = α/2
  • Resultant bisects the angle between the two forces
  • If α = 120°, then R = F
• If forces act along the same line and in the same direction (α = 0°):
  • R = F1 + F2
  • The resultant's direction is the same as the forces' line of action.
  • R represents the greatest or maximum value of the resultant.
• If forces act along the same line but in opposite directions (α = 180°):
  • R = |F1 - F2|
  • The resultant's direction matches the greater force's direction.
  • R is the smallest or minimum value of the resultant.
• If two forces are equal in magnitude but opposite in direction:
  • R = zero, resulting in equilibrium.
• If the resultant is perpendicular to the first force (θ = 90°):
  • R² = F2² - F1² (Pythagoras' theorem applications)
  • cos α = -F1 / F2 (α is an obtuse angle, where F1 < F2)
  • When the resultant is perpendicular to one of the two forces meaning it is perpendicular to smallest force

Resolution of a Force

• Resolving a force into components:
  • F1 / sin θ2 = F2 / sin θ1 = R / sin(θ1 + θ2)
  • F1 = (R sin θ2) / sin(θ1 + θ2) is the magnitude of R's component inclining by θ1 on R.
  • F2 = (R sin θ1) / sin(θ1 + θ2) is the magnitude of R's component inclining by θ2 on R.
• Special Cases:
  • Component F2 inclines at an angle θ to the direction of R:
    • F2 = R cos θ, F1 = R sin θ
  • Angle between R and the first component equals zero:
    • F1 = R, F2 = zero
  • i and j represent two perpendicular unit vectors along OX and OY, respectively, with O as the origin:
    • F1 = (R cos θ)î vector, F2 = (R sin θ)ĵ vector
    • R vector = F1 vector + F2 vector = (R cos θ)î vector + (R sin θ)ĵ vector
  • Given F = (F, θ):
    • F vector = F cos θî vector + F sin θĵ vector
  • If θ is within the range 0 to π/2:
    • The magnitudes of the two components (R cos θ) and (R sin θ) are less than the magnitude of the force R.
• 0 < sin θ < 1, 0 < cos θ < 1

Additional Notes

• A body of weight (w) on a smooth inclined plane at an angle (θ):
  • F1 = w sin θ (component along the greatest slope direction)
  • F2 = w cos θ (component perpendicular to the plane)

Lami's Rule

• If three coplanar forces meet at a point and are in equilibrium, each force’s magnitude is proportional to the sine of the angle between the other two forces.
  • F1 / sin θ1 = F2 / sin θ2 = F3 / sin θ3

Triangle of Forces

• For a rigid body in equilibrium under three forces, a triangle is drawn with sides parallel to the forces' lines of action taken in cyclic order.The sides' lengths are proportional to the corresponding forces' magnitudes.
• If the force triangle is ΔXYZ:
• F1 / XY = F2 / YZ = F3 / XZ

Equilibrium of Co-planar Forces at a Point

• If a rigid body is in conditions with three nonparallel forces, their lines of action meet at a single point.
• For a uniform rod with weight (w) in equilibrium against smooth vertical and rough horizontal ground:
  • The rod’s weight acts downwards at its midpoint (center of gravity).
  • The smooth wall's reaction (r1) is perpendicular to the wall (direction BD).
  • The rough ground's reaction (r2) has an unknown direction, determined by drawing AD through the intersection point D of w and r1.