Problem 77 - Master Resource Book in JEE Main Physics

Question

A particle of mass \(=5\) is moving with a uniform speed \(v=3 \sqrt{2}\) in the \(X O Y\) plane along the line \(y=x+4\). The magnitude of the angular momentum of the particle about the origin is (a) 60 units (b) \(40 \sqrt{2}\) units (c) \(7.5\) units (d) zero

Step-by-Step Solution

Answer

The magnitude of the angular momentum is 60 units.

Step 1: Understand the Concepts

Angular momentum \( L \) of a particle moving in a plane about a point can be calculated using the formula: \( L = m(v \cdot d) \), where \( m \) is mass, \( v \) is velocity, and \( d \) is the perpendicular distance from the point to the line of motion of the particle.

Step 2: Identify the Information Given

The mass \( m \) of the particle is 5, and the velocity \( v \) is \( 3 \sqrt{2} \). The line of motion is given by the equation \( y = x + 4 \). We need to determine the perpendicular distance from the origin \((0,0)\) to this line.

Step 3: Calculate Perpendicular Distance from Origin to the Line

The formula for the distance \( d \) from a point \((x_1, y_1)\) to a line \( Ax + By + C = 0 \) is \( d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \). Here, the line is \( x - y + 4 = 0 \) (since \( y = x + 4 \) rearranges to \( x - y + 4 = 0 \)), and the point is \( (0,0) \). Substitute these values into the formula: \( d = \frac{|0 - 0 + 4|}{\sqrt{1^2 + (-1)^2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \).

Step 4: Calculate the Angular Momentum

Using the formula \( L = m(v \cdot d) \), substitute the given values: \( L = 5 \times 3\sqrt{2} \times 2\sqrt{2} = 5 \times 3 \times 4 = 60 \). Thus, the magnitude of the angular momentum is 60 units.

Key Concepts

Physics Problems

Physics problems often involve several steps of calculation and reasoning, similar to solving an intricate puzzle. They test not just your analytical skills, but also your understanding of fundamental concepts. In this exercise, we are tasked with finding the angular momentum of a particle about a point. Angular momentum is a measure of the quantity of rotation of an object about an axis. It depends on three factors: the mass of the object, its velocity, and the perpendicular distance from the point of rotation to the line of motion.The challenge with physics problems is to correctly identify and apply these principles. Here, the mass of the particle is given as 5 units, and its velocity as 3\( \sqrt{2} \) units. Understanding how to extract and utilize these values is crucial. The velocity indicates how fast the particle is moving in a straight line in the plane, while the mass is a measure of its inertia.To find the angular momentum,

• First, calculate the perpendicular distance from the pivot point (here, the origin) to the particle's path.
• Next, use that distance along with the mass and velocity to compute the angular momentum.

By breaking the problem down into manageable parts, you make it easier to solve and gain insights into the movement and interactions of physical objects.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebraic equations to describe geometric principles and provide a powerful tool for solving a wide variety of problems. The particle in our problem moves along a line described by the equation \( y = x + 4 \). This linear equation tells us how the particle is positioned in the plane.To calculate the angular momentum, we must determine how this line relates to the origin. This is done by finding the perpendicular distance from the origin to the line, which is a common task in coordinate geometry. The equation of the line can be rearranged to \( x - y + 4 = 0 \), and using the distance formula\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]helps us find that distance. This conversion from line form to the equation form allows for simple calculations and is a fundamental use of coordinate geometry in solving such problems.By mastering these coordinate geometry techniques, you gain

• The ability to solve geometric problems algebraically.
• A deeper understanding of how lines and curves interact in a plane.

This solid foundation aids in tackling even more complex physics problems.

JEE Preparation

Preparing for the Joint Entrance Examination (JEE) involves a strategic approach focusing on understanding core concepts deeply and practicing a variety of problems. Physics, being one of the core subjects, demands not just rote memorization but a true grasp of concepts like angular momentum. To excel in the JEE, it is crucial to focus on:

• Developing a clear understanding of both basic and advanced physics concepts.
• Practicing a diverse range of problems, ranging from simple to complex, ensuring you are ready for any kind of twist in the examination.
• Timing your solution process to enhance speed without sacrificing accuracy.

Working through exercises such as calculating the angular momentum of a moving particle helps reinforce core principles and improve problem-solving skills. This example strengthens knowledge on applying formulas, understanding geometric-plane relationships, and integrating concepts to derive solutions. Consistent practice, coupled with review and analysis of both successes and errors, builds the comprehensive skill set required to succeed in JEE physics sections.