Motion in a Plane

Class 11 NCERT Physics

NCERT

1   State, for each of the following physical quantities, if it is a scalar or a vector:$\\$ volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.

Solution :

$\textbf{Scalar}:$ Volume, mass, speed, density, number of moles, angular frequency$\\$ $\textbf{Vector}:$ Acceleration, velocity, displacement, angular velocity$\\$ A scalar quantity is specified by its magnitude only. It does not have any direction associated with it.$\\$ Volume, mass, speed, density, number of moles, and angular frequency are some of the scalar physical quantities.$\\$ A vector quantity is specified by its magnitude as well as the direction associated with it.$\\$ Acceleration, velocity, displacement, and angular velocity belong to this category.

2   Pick out the two scalar quantities in the following list:$\\$ Force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.

Solution :

$\textbf{Work}$ and $\textbf{Current}$ are scalar quantities.$\\$ $\text{Work}$ done is given by the dot product of force and displacement. Since the dot product of two quantities is always a scalar, work is a scalar physical quantity.$\\$ $\text{Current}$ is described only by its magnitude. Its direction is not taken into account. Hence, it is a scalar quantity.

3   Pick out the only vector quantity in the following list:$\\$ Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.

Solution :

$\text{Impulse}$$\\$ $\text{Impulse}$ is given by the product of force and time. Since force is a vector quantity, its product with time (a scalar quantity) gives a vector quantity.

4   State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful:$\\$ (a) adding any two scalars,$\\$ (b) adding a scalar to a vector of the same dimension s,$\\$ (c) multiplying any vector by any scalar,$\\$ (d) multiplying any two scalars,$\\$ (e) adding any two vectors,$\\$ (f) adding a component of a vector to the same vector.

Solution :

(a) Not meaningful. The addition of two scalar quantities is meaningful only if they both represent the same physical quantity.$\\$ (b) Not meaningful. The addition of a vector quantity with a scalar quantity is not meaningful.$\\$ (c) Meaningful. A scalar can be multiplied with a vector. For example, force is multiplied with time to give impulse.$\\$ (d) Meaningful. A scalar, irrespective of the physical quantity it represents, can be multiplied with another scalar having the same or different dimensions.$\\$ (e) Not meaningful. The addition of two vector quantities is meaningful only if they both represent the same physical quantity.$\\$ (f) Meaningful A component of a vector can be added to the same vector as they both have the same dimensions.

5   Read each statement below carefully and state with reasons, if it is true or false:$\\$ (a) The magnitude of a vector is always a scalar,$\\$ (b) each component of a vector is always a scalar,$\\$ (c) the total path length is always equal to the magnitude of the displacement vector of a particle.$\\$ (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time,$\\$ (e) Three vectors not lying in a plane can never add up to give a null vector.

Solution :

(a) True. The magnitude of a vector is a number. Hence, it is a scalar.$\\$ (b) False. Each component of a vector is also a vector.$\\$ (c) False. Total path length is a scalar quantity, whereas displacement is a vector quantity. Hence, the total path length is always greater than the magnitude of displacement. It becomes equal to the magnitude of displacement only when a particle is moving in a straight line.$\\$ (d) True. It is because of the fact that the total path length is always greater than or equal to the magnitude of displacement of a particle.$\\$ (e) True. Three vectors, which do not lie in a plane, cannot be represented by the sides of a triangle taken in the same order.