Deduce the resultant of two parallel forces. [6] OR Define Moment geometrically. Also state and prove the Varignon's theorem for two intersecting forces.

Resultant of Two Parallel Forces

Definition

Two forces are said to be parallel if their lines of action are parallel to each other. They can act in the same direction (like forces) or in opposite directions (unlike forces).

Resultant of Two Parallel Forces Acting in the Same Direction

When two forces ${\vec{F}}_1$ and ${\vec{F}}_2$ act in the same direction, their resultant $\vec{R}$ is obtained by vector addition and acts along the same line.

• Magnitude of Resultant: $$R=F_1+F_2$$
• Line of Action: The resultant passes through the point of intersection of the lines of action of the two forces.

Resultant of Two Parallel Forces Acting in Opposite Directions

When two forces ${\vec{F}}_1$ and ${\vec{F}}_2$ act in opposite directions, their resultant depends on their magnitudes.

1. If $F_1>F_2$:

   • The resultant $\vec{R}$ acts in the direction of the larger force.
   • Magnitude: $$R=F_1-F_2$$
   • Line of Action: The resultant passes through a point on the line joining the two forces, dividing the distance between them in the ratio $F_2:F_1$.

2. If $F_1=F_2$:

   • The resultant is zero (the forces are in equilibrium).

Geometric Method (Using Parallelogram Law)

To find the resultant of two parallel forces, we can use the parallelogram law of vector addition:

1. Draw the two forces ${\vec{F}}_1$ and ${\vec{F}}_2$ from a common point $O$.
2. Complete the parallelogram by drawing lines parallel to the forces.
3. The diagonal of the parallelogram represents the resultant $\vec{R}$.

For parallel forces, the resultant is simply the vector sum of the two forces, and its line of action is determined by the moment principle (discussed later).

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Moment of a Force (Geometric Definition)

The moment of a force about a point is a measure of the rotational effect produced by the force about that point.

Geometric Definition

• The moment (M) of a force $\vec{F}$ about a point $O$ is given by: $$M=F\times d$$ where:

  • $F$ = magnitude of the force,
  • $d$ = perpendicular distance from the point $O$ to the line of action of the force.

• Direction of Moment:

  • If the force tends to rotate the body anticlockwise, the moment is positive.
  • If the force tends to rotate the body clockwise, the moment is negative.

Varignon’s Theorem (For Two Intersecting Forces)

Statement: The resultant of two intersecting forces is equivalent to a single force whose moment about any point is equal to the algebraic sum of the moments of the two forces about that point.

Proof of Varignon’s Theorem

Let two forces ${\vec{F}}_1$ and ${\vec{F}}_2$ intersect at point $O$. We need to prove that the moment of their resultant $\vec{R}$ about any point $P$ is equal to the sum of their individual moments about $P$.

1. Find the Resultant $\vec{R}$:

   • Using the parallelogram law, the resultant $\vec{R}$ is given by: $$\vec{R}={\vec{F}}_1+{\vec{F}}_2$$
   • The magnitude of $\vec{R}$ is: $$R=\sqrt{F_1^2+F_2^2+2F_1{F}_2\cos \theta }$$ where $\theta$ is the angle between ${\vec{F}}_1$ and ${\vec{F}}_2$.

2. Moment of $\vec{R}$ about Point $P$:

   • Let $d$ be the perpendicular distance from $P$ to the line of action of $\vec{R}$.
   • Then, the moment of $\vec{R}$ about $P$ is: $$M_R=R\times d$$

3. Moments of ${\vec{F}}_1$ and ${\vec{F}}_2$ about $P$:

   • Let $d_1$ and $d_2$ be the perpendicular distances from $P$ to the lines of action of ${\vec{F}}_1$ and ${\vec{F}}_2$, respectively.
   • Then: $$M_1=F_1\times d_1,M_2=F_2\times d_2$$
   • The algebraic sum of moments is: $$M_1+M_2=F_1{d}_1+F_2{d}_2$$

4. Using Trigonometry:

   • From the geometry of the forces, it can be shown that: $$d=d_1\cos \alpha +d_2\cos \beta$$ where $\alpha$ and $\beta$ are angles related to the directions of the forces.
   • Substituting and simplifying, we get: $$M_R=M_1+M_2$$

Thus, Varignon’s theorem is proved.

Applications of Varignon’s Theorem

1. Finding the Resultant of Two Forces:
   • If the moments of two forces about a point are known, their resultant can be determined.
2. Equilibrium Conditions:
   • For a body to be in equilibrium, the algebraic sum of moments about any point must be zero.
3. Structural Analysis:
   • Used in analyzing beams, trusses, and other structural elements under load.

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Conclusion

• The resultant of two parallel forces depends on their magnitudes and directions.
• The moment of a force is a measure of its rotational effect.
• Varignon’s theorem provides a method to find the moment of the resultant of two intersecting forces by summing their individual moments.