Problem 344 - JEE Physics

Question

Three Forces \(F_{1}, F_{2}\), and \(F_{3}\) together keep a body in equilibrium. If \(F_{1}=3 \mathrm{~N}\) along the positive \(\mathrm{X}\) - axis, \(\mathrm{F}_{2}=4 \mathrm{~N}\) along the positive Y-axis then the third force \(F_{3}\) is (A) \(5 \mathrm{~N}\) -making an angle \(\theta=\tan ^{-1}(3 / 4)\) with negative \(\mathrm{y}\) -axis (B) \(5 \mathrm{~N}\) -making an angle \(\theta=\tan ^{-1}(4 / 3)\) with negative \(\mathrm{y}\) -axis (C) \(7 \mathrm{~N}\) -making an angle \(\theta=\tan ^{-1}(3 / 4)\) with negative \(\mathrm{y}\) -axis (D) \(7 \mathrm{~N}\) -making an angle \(\theta=\tan ^{-1}(4 / 3)\) with negative \(\mathrm{y}\) -axis

Step-by-Step Solution

Answer

The third force \(F_{3}\) is 5 N, making an angle \(\theta = \tan^{-1}(3/4)\) with the negative Y-axis. Therefore, the correct answer is (A).

Step 1: Write down the given forces

We are given 3 forces acting on the body: \( F_1 = 3 \, N \) along the positive X-axis, \( F_2 = 4 \, N \) along the positive Y-axis, and \( F_3 \) that will balance out the other two forces.

Step 2: Find the X and Y components of the third force

For the body to be in equilibrium, the net force along both X and Y axis should be zero. So the X and Y components of F3 should cancel out with the given forces. Along X-axis, \(F_{3x} = - F_{1} = -3 \, N\) Along Y-axis, \(F_{3y} = - F_{2} = -4 \, N\)

Step 3: Calculate the magnitude and angle of F3

Now we can find the magnitude of the third force using the Pythagorean theorem: \(F_{3} = \sqrt{F_{3x}^2 + F_{3y}^2} = \sqrt{(-3)^2 + (-4)^2} = 5\, N\) We find the angle F3 makes with the negative Y-axis using the tangent function: \(\theta = \tan^{-1}\left(\frac{F_{3x}}{-F_{3y}}\right) = \tan^{-1}\left(\frac{-3}{-4}\right) = \tan^{-1}\left(\frac{3}{4}\right)\) Therefore, the third force F3 is 5 N, making an angle \(\theta = \tan^{-1}(3/4)\) with the negative Y-axis. This corresponds to option (A).

Key Concepts

Vector Components

In physics, understanding vector components is crucial when dealing with forces acting in multiple directions. A vector is a quantity that has both magnitude and direction. Real-life scenarios, like a force acting on an object, often require breaking them down into simpler parts to solve problems conveniently. These parts are known as components.
A vector acting in two dimensions can be resolved into two perpendicular components. Typically, these are along the X-axis (horizontal) and Y-axis (vertical). Knowing the components helps in simplifying rigorous vector calculations.
To find the components of a vector:

• The X-component represents the part of the vector that influences motion or force along the horizontal axis.
• The Y-component accounts for the influence along the vertical axis.

For example, if you have a force vector pointing diagonally in an XY-plane, you can break it down into a horizontal vector along the X-axis and a vertical vector along the Y-axis. This way, you can assess the impact of the force along each individual axis.
This process is particularly vital when solving equilibria problems since any imbalance in one direction can lead to motion or instability.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics used widely in physics, especially when working with vectors in 2D space. It relates the sides of a right-angle triangle, stating that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Mathematically, this is written as: \[ c^2 = a^2 + b^2 \]
where:

• \( c \) is the hypotenuse,
• \( a \) and \( b \) are the other two sides.

When applied to vectors, imagine you have an X-component and a Y-component forming a right-angle triangle with the vector being the hypotenuse.In the case of the problem, the force \( F_3 \) need balancing the two known forces. To achieve equilibrium in forces, the magnitude of the third force can be calculated using the components: \[ F_3 = \sqrt{F_{3x}^2 + F_{3y}^2} \] This allows you to find the actual magnitude of the resultant force that ensures the physical system remains balanced. Using this principle brings precision to calculations, aiding in deducing unknown parameters in equilibrium problems.

Trigonometric Functions

Trigonometric functions are a set of mathematical functions that relate angles of a triangle to the lengths of its sides. These are especially useful in evaluating angles and side ratios in right-angle triangles, which frequently appear in vector problems.
Common trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each serves specific purposes:

• Tangent of an angle is the ratio of the opposite side to the adjacent side (\( \tan \theta = \frac{opposite}{adjacent} \)).
• Sine involves the ratio of the opposite side to the hypotenuse, and cosine is the adjacent side over the hypotenuse.

For vectors, these functions help determine the directions of an unknown force relative to known components. In this case, trigonometric functions assist in finding the angle \( \theta \) that vector \( F_3 \) makes with a given axis.
You utilize the tangent function to determine the angle:\[ \theta = \tan^{-1} \left( \frac{F_{3x}}{-F_{3y}} \right) \] Here, \( F_{3x} \) and \( -F_{3y} \) are the components, and the resulting angle is measured with respect to the negative Y-axis. These functions are incredibly powerful for exactly aligning vectors in specific directions, which is often needed for understanding and solving equilibrium of force problems.