Problem 79 - Master Resource Book in JEE Main Physics

Question

There are two forces each of magnitude 10 units. One inclined at an angle of \(30^{\circ}\) and the other at an angle of \(135^{\circ}\) to the positive direction of \(x\)-axis. The \(x\) and \(y\) components of the resultant are respectively, (a) \(1.59 \hat{\mathrm{i}}\) and \(12.07 \hat{\mathrm{j}}\) (b) \(10 \hat{i}\) and \(10 \hat{\mathrm{j}}\) (c) \(1.59 \hat{\mathrm{i}}\) (d) \(15.9 \hat{\mathrm{i}}\) and \(12.07 \hat{\mathrm{j}}\)

Step-by-Step Solution

Answer

Option (a) is correct: \(1.59 \hat{\mathrm{i}}\) and \(12.07 \hat{\mathrm{j}}\).

Step 1: Resolve the First Force

The first force has a magnitude of 10 units and is inclined at an angle of \(30^{\circ}\) with the positive x-axis. The components of this force are calculated as: - **x-component:** \( F_{1x} = 10 \cos(30^{\circ}) = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66 \)- **y-component:** \( F_{1y} = 10 \sin(30^{\circ}) = 10 \times \frac{1}{2} = 5 \)

Step 2: Resolve the Second Force

The second force also has a magnitude of 10 units and is inclined at an angle of \(135^{\circ}\) to the positive x-axis. The components of this force are calculated as:- **x-component:** \( F_{2x} = 10 \cos(135^{\circ}) = 10 \times \frac{-1}{\sqrt{2}} = -5\sqrt{2} \approx -7.07 \)- **y-component:** \( F_{2y} = 10 \sin(135^{\circ}) = 10 \times \frac{1}{\sqrt{2}} = 5\sqrt{2} \approx 7.07 \)

Step 3: Calculate the Resultant Force Components

Sum up the components from both forces to determine the components of the resultant force.- **Resultant x-component:** \( R_{x} = F_{1x} + F_{2x} = 8.66 + (-7.07) = 1.59 \)- **Resultant y-component:** \( R_{y} = F_{1y} + F_{2y} = 5 + 7.07 = 12.07 \)

Step 4: Select the Correct Answer

The resultant components calculated are \(1.59 \hat{\mathrm{i}} \) and \( 12.07 \hat{\mathrm{j}} \). Comparing this with the options given:Option (a) "\(1.59 \hat{\mathrm{i}}\) and \(12.07 \hat{\mathrm{j}}\)" is the correct choice.

Key Concepts

Vector Components

Understanding vector components is crucial in physics, particularly when dealing with forces. Vectors are quantities that have both magnitude and direction. When we resolve a vector into components, we break it down into two perpendicular directions, usually the x and y axes. Imagine you're looking at a force acting diagonally - by finding its components, you can see how much of that force is pushing sideways and how much is pushing upwards. For any vector, the magnitude and angle determine its x and y components. The formula for resolving these components involves the trigonometric functions cosine and sine, respectively. This step is vital because it turns a single vector into two simpler, manageable parts that are easier to work with. Consequently, finding vector components is the first step in many physics problems, including determining the resultant force from multiple vectors.

Trigonometry in Physics

Trigonometry is the mathematics of triangles, and it plays a critical role in physics, especially when analyzing vectors. When a force acts at an angle, trigonometry helps us break it down into more straightforward components along the x and y axes. The angle of a force with respect to a reference axis, usually the x-axis, is fundamental in computing these components.

The basic trigonometric functions used are:

• Cosine: Used to find the adjacent side of a right-angled triangle, or in vector terms, the horizontal or x-component. For an angle \(\theta\), the x-component is given by \(F \cos \theta\).
• Sine: Presented to discover the side opposite the angle, or the vertical or y-component, expressed as \(F \sin \theta\).

Through these calculations, complex force vectors are simplified, making it possible to comprehend and manipulate them in problem-solving contexts. Hence, a good grasp of trigonometry is indispensable for anyone studying physics.

Force Resolution

Force resolution is the process of breaking down a single force into multiple components, typically along the x and y axes in a two-dimensional space. This technique simplifies the analysis of forces acting in various directions, as it allows us to consider each axis or direction independently.

A typical method to resolve forces is to start by identifying the angle at which the force acts concerning the axes. Following that, use trigonometric functions to find the horizontal and vertical components. For instance, a force of 10 units at \(30^{\circ}\) yields an x-component of \(10 \cos 30^{\circ} \approx 8.66\) and a y-component of \(10 \sin 30^{\circ} \approx 5\).

Resolving forces into components is invaluable because the effects of the force in each direction can be individually analyzed and combined with other forces' effects in those directions. This is why correctly resolving forces is key to accurately predicting the resultant force, ensuring precise solutions to physical problems.