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

The classification of the given physical quantities into scalar and vector is as follows:

• Scalar Quantities: Volume, mass, speed, density, number of moles, angular frequency.
• Vector Quantities: Acceleration, velocity, displacement, angular velocity.

Reasoning:

• A scalar quantity is completely described by its magnitude alone. It does not possess any direction. Examples from the list include volume, mass, speed, density, number of moles, and angular frequency.
• A vector quantity requires both magnitude and direction for its complete description. Examples from the list include acceleration, velocity, displacement, and angular velocity.

The two scalar quantities from the given list are work and current.

Explanation:

• 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.
• Current is considered a scalar quantity in this context because it is typically defined by its magnitude (e.g., 5 Amperes). Although current has a direction of flow, it does not follow the vector addition rules, which is a characteristic of vector quantities.

The only vector quantity in the given list is impulse.

Explanation:

• Impulse is defined as the product of force and the time interval over which the force acts ($Impulse\; =\; Force\; \backslash times\; \backslash Delta\; t$). Since force is a vector quantity and time interval ($\backslash Delta\; t$) is a scalar quantity, their product (impulse) is a vector quantity. Impulse has both magnitude and direction, typically in the direction of the force.
• 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 operation is meaningful only if the two scalars represent the same physical quantity (e.g., adding mass to mass, or speed to speed). Adding scalars of different physical quantities (e.g., mass + speed) is not 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 these fundamentally different types of quantities does not yield a physically meaningful result.
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 does not change its fundamental nature as a vector. For example, multiplying force (vector) by time (scalar) gives impulse (vector).
4. Multiplying any two scalars: Meaningful. This operation is always meaningful, regardless of whether the scalars represent the same or different physical quantities. The result is another scalar quantity.
5. Adding any two vectors: Meaningful. This operation is meaningful only if the two vectors represent the same physical quantity (e.g., adding two forces, or two velocities). The result is a vector quantity.
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, such as resolving a vector into its components or performing vector addition using components.

Let's evaluate each statement:

1. True. The magnitude of a vector represents its size or length and is a quantity that has only magnitude, thus it is a scalar.
2. False. Each component of a vector (e.g., the x-component, y-component, or z-component) is a scalar quantity. For example, if a vector is $\backslash vec\{A\}\; =\; A\_x\; \backslash hat\{i\}\; +\; A\_y\; \backslash hat\{j\}$, then $A\_x$ and $A\_y$ are scalars representing the magnitudes of the vector along the respective axes.
3. False. The total path length is equal to the magnitude of the displacement vector only if the particle moves in a straight line without reversing its direction. In general, the path length is greater than the magnitude of the displacement. For example, if a particle moves from point A to point B and then back to A, the displacement is zero, but the path length is non-zero.
4. True. Average speed is defined as $\backslash text\{Average\; Speed\}\; =\; \backslash frac\{\backslash text\{Total\; Path\; Length\}\}\{\backslash text\{Time\; Interval\}\}$. The magnitude of average velocity is defined as $|\backslash vec\{v\}\_\{avg\}|\; =\; \backslash frac\{|\backslash text\{Displacement\}|\}\{\backslash text\{Time\; Interval\}\}$. Since the total path length is always greater than or equal to the magnitude of the displacement, the average speed is always greater than or equal to the magnitude of the average velocity.
5. True. If three vectors add up to give a null vector ($\backslash vec\{0\}$), it means they can form a closed triangle (if coplanar) or a closed polygon in 3D space. If three vectors are not coplanar, they can still add up to a null vector. For example, consider three vectors representing the edges of a tetrahedron originating from a common vertex. If these vectors are $\backslash vec\{a\}$, $\backslash vec\{b\}$, and $\backslash vec\{c\}$, and the fourth vertex is at the origin, then the vector sum $\backslash vec\{a\}\; +\; \backslash vec\{b\}\; +\; \backslash vec\{c\}$ does not necessarily result in a null vector. However, it is possible to have three non-coplanar vectors that sum to zero. For instance, consider vectors along the edges of a non-planar tetrahedron. The statement implies that if you have three vectors that do not lie in the same plane, they cannot possibly cancel each other out to result in zero. This is incorrect. For example, consider vectors $\backslash vec\{a\}\; =\; (1,0,0)$, $\backslash vec\{b\}\; =\; (0,1,0)$, and $\backslash vec\{c\}\; =\; (-1,-1,0)$. These are coplanar. However, if we consider $\backslash vec\{a\}=(1,0,0)$, $\backslash vec\{b\}=(0,1,0)$, and $\backslash vec\{c\}=(-1,-1,1)$, these are not coplanar and their sum is $(0,0,1)$, not null. The statement is actually true in the sense that if three vectors are not coplanar, they cannot form a closed polygon in 3D space that sums to zero unless they are specifically chosen to do so. The statement is often interpreted in the context of forming a closed geometrical figure. If three vectors sum to zero, they can be represented as sides of a triangle. If they are not coplanar, they cannot form a triangle. Therefore, three non-coplanar vectors cannot add up to a null vector. This statement is true.