Problem 6 - Master Resource Book in JEE Main Physics

Question

The sum of the magnitudes of two forces acting at a point is \(16 \mathrm{~N}\). The resultant of these forces is perpendicular to the smaller force has a magnitude of \(8 \mathrm{~N}\). If the smaller force is magnitude \(x\), then, the value of \(x\) is (a) \(2 \mathrm{~N}\) (b) \(4 \mathrm{~N}\) (c) \(6 \mathrm{~N}\) (d) \(7 \mathrm{~N}\)

Step-by-Step Solution

Answer

The magnitude of the smaller force is 8 N.

Step 1: Understanding the Problem

We are given two forces, one with magnitude \( x \) (smaller force) and another with magnitude \( 16 - x \). The resultant force of magnitude 8 N is perpendicular to the smaller force. We need to find the value of \( x \).

Step 2: Applying the Perpendicularity Condition

When the resultant force is perpendicular to one of the original forces, the magnitude of the resultant is equal to the magnitude of the other original force. Here, \( R = 8 \text{ N} \) is perpendicular to the force \( x \text{ N} \), therefore, the magnitude of the other force, \( 16 - x \), must equal 8 N.

Step 3: Setting up the Equation

The equation arises from the perpendicularity condition: \[ 16 - x = 8 \]

Step 4: Solving the Equation

Solve the equation for \( x \):\[ 16 - x = 8 \]Subtract 16 from both sides:\[ -x = -8 \]Multiply or divide both sides by -1:\[ x = 8 \]

Step 5: Verifying the Solution

Check the solution by ensuring that the conditions of the problem are satisfied. With \( x = 8 \), the force magnitudes are \( 8 \text{ N} \) and \( 16 - 8 = 8 \text{ N} \). Their sum is 16 N, and the resultant is perpendicular to the smaller force; thus, the magnitude of the resultant is also 8 N. The conditions match, so \( x = 8 \text{ N} \) is confirmed.

Key Concepts

Resultant Force

In physics, the resultant force is essentially one single force that represents the action of multiple forces acting at a point. When multiple forces are applied to an object, they can collectively influence motion. They may not act straight along one line, and that's where the concept of the resultant force becomes imperative.

To find a resultant force, you need to consider both the magnitudes and the directions of the forces involved. When these forces act at a point, they can be simplified into a single vector that gives the same overall effect.

• The direction of the resultant force helps in understanding the overall impact on movement or motion.
• Its magnitude can be computed through vector addition methodologies like the triangle or parallelogram law.

By breaking forces into components and using basic trigonometry, you can solve for the resultant force quite effectively. This aids in a wide range of applications from mechanical systems to natural phenomena.

Perpendicular Vectors

Perpendicular vectors are a fascinating concept in vector analysis. Two vectors are said to be perpendicular when the angle between them is exactly 90 degrees. This is not just a geometric idea but holds significant implications in physics and engineering.

One key property of perpendicular vectors is that their dot product is zero. In the context of forces, when a resultant is perpendicular to one of the forces, the entire dynamic of the vector system changes. It implies a balance of forces.

• The perpendicularity ensures minimal interaction in specific directions, which can be crucial for certain engineering designs.
• By understanding this relationship, we can simplify complex systems and analyze them with ease.

In practical applications, ensuring forces are perpendicular can help optimize designs, such as in structures and mechanical systems, to manage loads effectively and prevent undesired stress.

Magnitude of Forces

The magnitude of a force is a measure of its strength and the extent to which it can change the motion of an object. In vector physics, magnitude is just another term for the size or length of the vector without considering its direction.

Understanding the magnitude is key in solving various physics problems, as it helps to quantify the force's impact. The calculation often involves breaking forces into their respective components and using the Pythagorean theorem.

• If two vectors are combined, the resultant's magnitude depends heavily on both the size and the angle between them.
• Being aware of how these factors interact allows us to predict motion and design stable structures efficiently.

Grasping the magnitude of forces ensures that we can predict the kinetic effects more accurately, thereby enabling us to manipulate physical systems to our desired outcome.