Problem 25 - Master Resource Book in JEE Main Physics

Question

The radius vector and linear momentum are respectively given by vector \(2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\). Their angular momentum is (a) \(2 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}\) (b) \(4 \hat{i}-8 \hat{k}\) (c) \(2 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\) (d) \(4 \hat{i}-8 \hat{j}\)

Step-by-Step Solution

Answer

The correct answer is (b) \(4 \hat{i}-8 \hat{k}\).

Step 1: Identify the Given Vectors

The problem provides the radius vector \( \mathbf{r} \) as \( 2 \hat{\mathbf{i}} + 2 \hat{\mathbf{j}} + \hat{\mathbf{k}} \) and the linear momentum vector \( \mathbf{p} \) as \( 2 \hat{\mathbf{i}} - 2 \hat{\mathbf{j}} + \hat{\mathbf{k}} \).

Step 2: Recall Angular Momentum Formula

The angular momentum \( \mathbf{L} \) of a particle is given by the cross product of the radius vector \( \mathbf{r} \) and the linear momentum \( \mathbf{p} \): \( \mathbf{L} = \mathbf{r} \times \mathbf{p} \).

Step 3: Set Up the Cross Product Calculation

The cross product \( \mathbf{r} \times \mathbf{p} \) can be found using the determinant:\[\mathbf{L} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ 2 & 2 & 1 \ 2 & -2 & 1 \end{vmatrix}\]

Step 4: Calculate the Determinant

Expand the determinant to get the cross product:\[\mathbf{L} = \hat{\mathbf{i}} \begin{vmatrix} 2 & 1 \ -2 & 1 \end{vmatrix} - \hat{\mathbf{j}} \begin{vmatrix} 2 & 1 \ 2 & 1 \end{vmatrix} + \hat{\mathbf{k}} \begin{vmatrix} 2 & 2 \ 2 & -2 \end{vmatrix} \]Calculate each minor determinant separately:

Step 5: Compute Each Minor Determinant

- \( \hat{\mathbf{i}} \): \( 2\cdot1 - (-2)\cdot1 = 2 + 2 = 4 \)- \( \hat{\mathbf{j}} \): \( 2\cdot1 - 2\cdot1 = 2 - 2 = 0 \)- \( \hat{\mathbf{k}} \): \( 2\cdot(-2) - 2\cdot2 = -4 - 4 = -8 \)

Step 6: Combine the Results

Combine the results from the minor determinants to form the final vector: \[ \mathbf{L} = 4 \hat{\mathbf{i}} - 0 \hat{\mathbf{j}} - 8 \hat{\mathbf{k}} = 4 \hat{\mathbf{i}} - 8 \hat{\mathbf{k}} \]

Step 7: Select the Correct Answer

The calculated angular momentum vector \( 4 \hat{\mathbf{i}} - 8 \hat{\mathbf{k}} \) corresponds to option (b): \(4 \hat{i}-8 \hat{k}\).

Key Concepts

Cross Product

In physics, the cross product is a method to calculate the angular momentum by combining two vectors: the radius vector and the linear momentum. The formula for the angular momentum \( \mathbf{L} \) is given by the equation \( \mathbf{L} = \mathbf{r} \times \mathbf{p} \), where \( \mathbf{r} \) is the radius vector and \( \mathbf{p} \) is the linear momentum vector.
This operation is typically represented as a determinant:

• The top row consists of unit vectors \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}} \).
• The second row contains the components of the radius vector.
• The third row contains the components of the linear momentum.

To determine the cross product, compute the determinant by expanding it:
You subtract the \( \hat{\mathbf{j}} \) component and add the \( \hat{\mathbf{k}} \) component, solving the respective 2x2 matrices. This results in a new vector, \( 4 \hat{\mathbf{i}} - 8 \hat{\mathbf{k}} \), which is the angular momentum of the object with respect to a given axis. This approach can visually be compared to calculating the area of a parallelogram generated by two vectors.

Linear Momentum

Linear momentum is a fundamental physics concept linked to the mass and velocity of an object. It is defined by the vector \( \mathbf{p} = m \mathbf{v} \), where \( m \) is mass and \( \mathbf{v} \) is velocity. In this exercise, however, we are given linear momentum directly as a vector: \(2 \hat{\mathbf{i}} - 2 \hat{\mathbf{j}} + \hat{\mathbf{k}}\).
Linear momentum signifies how much motion an object possesses. It is always directional, relying on the path of an object's movement.
This vector nature means the linear momentum indicates both the magnitude and direction of the object's velocity. The momentum vector plays a crucial role when calculating the angular momentum.
It's important to understand that while linear momentum deals with straight-line paths, angular momentum combines this concept with rotation around a point, offering a broader perspective of motion.

Radius Vector

The radius vector, often denoted as \( \mathbf{r} \), is the positional vector from a fixed point (usually the origin) to the current location of the object. Here, it is represented as \(2 \hat{\mathbf{i}} + 2 \hat{\mathbf{j}} + \hat{\mathbf{k}}\). This vector is crucial in physics, as it helps define the distance and direction from the origin point.
The radius vector is essential in determining angular momentum because it acts as the pivot point from which the object's rotational motion is calculated.
Essentially, it helps establish a reference frame, thereby laying out a path over which the forces are applied.
Understanding the radius vector allows us to see how far and in what direction the object is located from the origin, integrating a spatial perspective into the motion description. By identifying its components, \(x\), \(y\), and \(z\), it's easier to make cross product calculations with linear momentum, leading to a better understanding of rotational dynamics.