Problem 554 - JEE Physics

Question

Two bodies of mass \(1 \mathrm{~kg}\) and \(3 \mathrm{~kg}\) have position vector \((\mathrm{i} \wedge+2 \mathrm{j} \wedge+\mathrm{k} \wedge)\) and \((-3 \mathrm{i}-2 \mathrm{j}+\mathrm{k})\) respectively the centre of mass of this system has a position vector ..... \(\\{\mathrm{A}\\}-2 \mathrm{i} \wedge+2 \mathrm{k} \wedge\) \(\\{B\\}-2 i \wedge-j \wedge+k \wedge\) \(\\{C\\} 2 i \wedge-j \wedge-k \wedge\) \(\\{\mathrm{D}\\}-\mathrm{i} \wedge+\mathrm{j} \wedge+\mathrm{k} \wedge\)

Step-by-Step Solution

Answer

The position vector of the center of mass of this system is \(-2 i \wedge-j \wedge+k \wedge\).

Step 1: Write down the given information

We have: - Masses: \(m_1 = 1 \mathrm{~kg}\) and \(m_2 = 3 \mathrm{~kg}\) - Position vectors: \(\vec{r}_1 = (\mathrm{i} \wedge+2 \mathrm{j} \wedge+\mathrm{k} \wedge)\) and \(\vec{r}_2 = (-3\mathrm{i}-2 \mathrm{j}+\mathrm{k})\)

Step 2: Calculate the center of mass position vector

Using the formula for the center of mass position vector, we have: $$\vec{R}_{\mathrm{cm}} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2}{m_1 + m_2}$$ Substitute the given values into the formula: $$\vec{R}_{\mathrm{cm}} = \frac{1(\mathrm{i} \wedge+2 \mathrm{j} \wedge+\mathrm{k} \wedge) + 3(-3\mathrm{i}-2 \mathrm{j}+\mathrm{k})}{1+3}$$

Step 3: Simplify the position vector

Now, perform the necessary operations: $$\vec{R}_{\mathrm{cm}} = \frac{1\,\mathrm{i} + 2\,\mathrm{j} + \mathrm{k} - 9\,\mathrm{i} - 6\,\mathrm{j} + 3\,\mathrm{k}}{4}$$ Combine the like terms: $$\vec{R}_{\mathrm{cm}} = \frac{-8\,\mathrm{i} - 4\,\mathrm{j} + 4\,\mathrm{k}}{4}$$ Divide each term by the denominator (4): $$\vec{R}_{\mathrm{cm}} = -2\,\mathrm{i} - \mathrm{j} + \mathrm{k}$$

Step 4: Compare to the given options

We now compare our result to the given options: A) \(-2 \mathrm{i} \wedge+2 \mathrm{k} \wedge\) B) \(-2 i \wedge-j \wedge+k \wedge\) C) \(2 i \wedge-j \wedge-k \wedge\) D) \(-\mathrm{i} \wedge+\mathrm{j} \wedge+\mathrm{k} \wedge\) Our result matches option B: $$\vec{R}_{\mathrm{cm}} = -2 i \wedge-j \wedge+k \wedge$$ So, the position vector of the center of mass of this system is given by option B.

Key Concepts

Position Vector

A position vector is a vector that describes the location of a point in space relative to a chosen origin. It is represented by coordinates in a specific reference frame, such as \( \vec{r} = (x, y, z) \). Each coordinate corresponds to the displacement from the origin along the x, y, and z axes, respectively.
In the exercise, two position vectors are given for two objects:

• \( \vec{r}_1 = (\mathrm{i} + 2\mathrm{j} + \mathrm{k}) \)
• \( \vec{r}_2 = (-3\mathrm{i} - 2\mathrm{j} + \mathrm{k}) \)

These vectors indicate the positions of the masses 1 kg and 3 kg in space. The position vectors are essential in calculating the center of mass, as they provide the necessary coordinates to determine the mass-weighted average position of the system.

Mass Distribution

Mass distribution refers to how mass is spread out in an object or a system of objects. In physics, understanding mass distribution helps in determining the center of mass, a crucial point that effectively summarizes the entire system’s mass position.
In our exercise, the system consists of two point masses:

• A 1 kg mass located at \( \vec{r}_1 \)
• A 3 kg mass located at \( \vec{r}_2 \)

The position of the center of mass depends on both the locations and the masses. Because the 3 kg mass is larger than the 1 kg mass, its position influences the center of mass proportionally more.
This concept helps us use mass to determine how the position vector of the entire system balances out, ensuring that the system's mass is equitably represented in defining its overall position.

Vector Addition

Vector addition is a method used to combine vectors into a single resultant vector. This involves summing corresponding components of the vectors. For position vectors, this allows us to find effective positions by combining the magnitude and directions.
To find the position vector of the center of mass, we use the formula:\[\vec{R}_{\mathrm{cm}} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2}{m_1 + m_2}\] The steps include:

• Multiplying each mass by its respective position vector: \( 1(\mathrm{i} + 2\mathrm{j} + \mathrm{k}) + 3(-3\mathrm{i} - 2\mathrm{j} + \mathrm{k}) \)
• Adding the vectors by summing each component: \(1\mathrm{i} - 9\mathrm{i} - 2\mathrm{j} + 6\mathrm{j} + \mathrm{k} + 3\mathrm{k}\)
• Resulting in: \(-8\mathrm{i} - 4\mathrm{j} + 4\mathrm{k}\)
• Finally dividing by the total mass \(4\), giving \(\vec{R}_{\mathrm{cm}} = -2 \mathrm{i} - \mathrm{j} + \mathrm{k} \)

This shows how vector addition helps to calculate the position of a combined mass system.

Mechanics Problem-Solving

Mechanics problem-solving involves applying various physics principles to solve problems related to motion, forces, and their effects on matter. This exercise illustrates a classic method of determining the center of mass using known positions and masses.
To solve this problem efficiently, one must:

• Identify the given elements, such as masses and position vectors.
• Apply the center of mass formula, which uses these inputs to generate a solution.
• Simplify the calculation by performing accurate vector operations.

This method not only helps find the center of mass but also enhances our understanding of how physical systems are described through vectors and mass properties. Knowing the step-by-step process of simplifying vector expressions and verifying results teaches crucial analytical skills used in more complex mechanics scenarios.