Question

State the Law of Parallelogram of Forces.

Answer 1

The Law of Parallelogram of Forces

The law states that:

"If two forces, acting simultaneously on a particle, are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram drawn from that same point."

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Explanation in Detail

This law provides a graphical and mathematical method for finding the resultant force, which is the single force that has the same effect as the two original forces combined.

1. Key Concepts

• Forces as Vectors: Forces are vector quantities, meaning they have both a magnitude (how strong the force is) and a direction. They are represented by arrows, where the length of the arrow indicates the magnitude and the arrowhead points in the direction.
• Concurrent Forces: The law applies to forces that act on the same point at the same time.
• Resultant Force (R): The combined effect or sum of two or more forces.

2. Visual Representation

Imagine a point O where two forces, F₁ and F₂, are acting.

1. Represent the Forces: Draw a vector (arrow) OA from point O to represent force F₁. The length of OA is proportional to the magnitude of F₁, and its direction is the direction of F₁.
2. From the same point O, draw another vector OB to represent force F₂.
3. Complete the Parallelogram: Draw a line from point A parallel to OB, and another line from point B parallel to OA. These lines will meet at a point C, completing the parallelogram OACB.
4. Find the Resultant: Draw the diagonal of the parallelogram starting from the common point O to the opposite corner C. This diagonal, OC, represents the resultant force R.

• The magnitude of the resultant force R is represented by the length of the diagonal OC.
• The direction of the resultant force R is the direction of the diagonal OC.

3. Mathematical Formulation

Using trigonometry, we can derive formulas to calculate the exact magnitude and direction of the resultant force.

Let:
$F_1$ and $F_2$ be the magnitudes of the two forces.
$\theta$ (theta) be the angle between the two forces $F_1$ and $F_2$.
$R$ be the magnitude of the resultant force.
$\alpha$ (alpha) be the angle that the resultant force $R$ makes with the force $F_1$.

Magnitude of the Resultant (R):
The magnitude is found using the Law of Cosines.

$R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos(\theta)}$

Direction of the Resultant ($\alpha$):
The direction is found using the Law of Sines or basic trigonometry.

$\tan(\alpha) = \frac{F_2 \sin(\theta)}{F_1 + F_2 \cos(\theta)}$

Example

Imagine two people pulling a box. Person 1 pulls with a force of 40 Newtons ($F_1$) to the east. Person 2 pulls with a force of 30 Newtons ($F_2$) to the north. The angle ($\theta$) between them is 90°.

• Magnitude:
  $R = \sqrt{40^2 + 30^2 + 2(40)(30) \cos(90^\circ)}$
  Since $\cos(90^\circ) = 0$, the formula simplifies to:
  $R = \sqrt{1600 + 900 + 0} = \sqrt{2500} = 50$ Newtons.

• Direction:
  $\tan(\alpha) = \frac{30 \sin(90^\circ)}{40 + 30 \cos(90^\circ)}$
  Since $\sin(90^\circ) = 1$ and $\cos(90^\circ) = 0$:
  $\tan(\alpha) = \frac{30(1)}{40 + 0} = \frac{30}{40} = 0.75$
  $\alpha = \arctan(0.75) \approx 36.87^\circ$

So, the combined effect is a single force of 50 Newtons acting at an angle of 36.87° north of east.