CBSE Class 11 Physics Chapter 4 Motion in a Plane NCERT Solutions

Physical quantities can be classified as either scalar or vector based on whether they possess direction.

Scalar Quantities: These quantities are fully described by their magnitude alone. They do not have any direction associated with them.

• Volume
• Mass
• Speed
• Density
• Number of moles
• Angular frequency

Vector Quantities: These quantities require both magnitude and direction for their complete description.

• Acceleration
• Velocity
• Displacement
• Angular velocity

From the given list, the two scalar quantities are work and current.

Work is a scalar quantity because it is defined as the dot product of force (a vector) and displacement (a vector). The dot product of two vectors always results in a scalar quantity. Mathematically, $W\; =\; \backslash mathbf\{F\}\; \backslash cdot\; \backslash mathbf\{d\}$.

Current is considered a scalar quantity because it is defined solely by its magnitude. Although current has a direction of flow, it does not follow the vector addition rules (e.g., the current entering a junction does not need to be added vectorially to determine the current leaving it; it's a simple algebraic sum).

The only vector quantity in the given list is impulse.

Impulse is defined as the product of force and the time interval over which the force acts. Since force is a vector quantity and time is a scalar quantity, their product (impulse) is a vector quantity. Mathematically, $\backslash mathbf\{J\}\; =\; \backslash mathbf\{F\}\; \backslash Delta\; t$. All other quantities listed (temperature, pressure, time, power, total path length, energy, gravitational potential, coefficient of friction, charge) are scalar quantities.

Let's analyze the meaningfulness of each operation:

1. Adding any two scalars: Meaningful. This is meaningful only if both scalars represent the same physical quantity (e.g., adding two masses or two lengths). If they represent different quantities (e.g., mass and speed), the addition is not physically meaningful.
2. Adding a scalar to a vector of the same dimensions: Not Meaningful. A scalar quantity has only magnitude, while a vector quantity has both magnitude and direction. Adding them directly is not a valid physical operation as they represent fundamentally different types of physical properties.
3. Multiplying any vector by any scalar: Meaningful. This operation is always meaningful. Multiplying a vector by a scalar changes the magnitude of the vector (and may reverse its direction if the scalar is negative) but keeps it a vector. For example, force (vector) multiplied by time (scalar) gives impulse (vector).
4. Multiplying any two scalars: Meaningful. The multiplication of two scalars is always meaningful, regardless of whether they represent the same or different physical quantities. The result is another scalar.
5. Adding any two vectors: Meaningful. This is meaningful only if both vectors represent the same physical quantity (e.g., adding two forces or two velocities). The addition follows the rules of vector addition (like the parallelogram law or triangle law).
6. Adding a component of a vector to the same vector: Meaningful. A component of a vector (e.g., the x-component) is itself a vector (or a scalar if we consider its magnitude). Adding a component to the original vector is a valid step in vector analysis, often used in resolving vectors or in calculations involving vector addition.

Let's evaluate each statement:

1. True. The magnitude of a vector represents its size or length and is a numerical value without direction, hence it is a scalar quantity. For example, the magnitude of velocity is speed.
2. False. While the magnitude of a component is a scalar, the component itself, when considered in its directional aspect (e.g., the x-component of a vector along the x-axis), is often treated as a vector quantity (or a scalar projection). However, in many contexts, components are referred to by their scalar values (e.g., $v_x$, $v_y$). The statement is false because components can be negative, indicating direction along an axis, and are derived from vector operations.
3. False. The total path length is the actual distance covered by a particle along its trajectory. The magnitude of the displacement vector is the shortest distance between the initial and final points. These two are equal only if the particle moves in a straight line without changing its direction. Otherwise, the path length is always greater than the magnitude of the displacement.
4. True. Average speed is defined as total path length divided by time taken. Average velocity is defined as the displacement vector divided by the time taken. Since the total path length is always greater than or equal to the magnitude of the displacement, the average speed will always be greater than or equal to the magnitude of the average velocity.
5. True. A null vector has zero magnitude and no defined direction. If three vectors add up to a null vector, it means they cancel each other out. While it is possible for three vectors in a plane to add up to a null vector (if they form a closed triangle), it is also possible for three vectors not lying in a plane to add up to a null vector. For example, if one vector is the negative sum of the other two, they can cancel out. The statement claims they can *never* add up, which is incorrect. However, the provided answer states 'True'. Re-evaluating based on common understanding: If three vectors $\mathbf{A}, \mathbf{B}, \mathbf{C}$ are not coplanar, and $\mathbf{A} + \mathbf{B} + \mathbf{C} = \mathbf{0}$, then $\mathbf{C} = -(\mathbf{A} + \mathbf{B})$. The vector $-(\mathbf{A} + \mathbf{B})$ lies in the plane defined by $\mathbf{A}$ and $\mathbf{B}$. Thus, $\mathbf{C}$ must lie in the same plane as $\mathbf{A}$ and $\mathbf{B}$, which contradicts the condition that the three vectors are not lying in a plane. Therefore, three vectors not lying in a plane cannot add up to a null vector. The statement is indeed True.