# 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 :

Let two vectors a and b be represented by the adjacent sides of a parallelogram $OMNP$, as shown in the given figure

Here we can write $|\bar{OM}|=|\bar{a } \quad ...(i)\\ |\bar{MN}|=|\bar{OP}|=|\bar{b}|\quad ...(ii)\\ |\bar{ON}|=|\bar{a}+\bar{b}| \quad ...(iii)$$\\ In a triangle, each side is smaller than the sum of the other two sides. Therefore, in \Delta OMN , we have:\\ ON < ( OM + MN )\\ |\bar{a}+\bar{b}| < |\bar{a}|+|\bar{b}| \quad ....(iv)$$\\$ If the two vectors $\bar{ a}$ and $\bar{b}$ act along a straight line in the same direction, then we can write:$\\$ $|\bar{a}+\bar{b}|=|\bar{a}|+|\bar{b}| \quad ...(v)$$\\ Combining equations (iv) and (v), we get:\\ |\bar{a}+\bar{b}| \leq |\bar{a}|+|\bar{b}|$$\\$ Let two vectors $\bar{a}$ and $\bar{b}$ be represented by the adjacent sides of a parallelogram $OMNP$, as shown in the given figure.$\\$ Here,we have: $\\$ $|\bar{OM}|=|\bar{a } \quad ...(i)\\ |\bar{MN}|=|\bar{OP}|=|\bar{b}|\quad ...(ii)\\ |\bar{ON}|=|\bar{a}+\bar{b}| \quad ...(iii)$$\\ In a triangle, each side is smaller than the sum of the other two sides. Therefore, in \Delta OMN , we have:\\ ON+MN > OM\\ ON+ OM > MN\\ |ON| > |\bar{OM}-\bar{OM}| \quad (\therefore OP=MN)\\ |\bar{a}+\bar{b}| > ||\bar{a}|-|\bar{b}|| \quad (iv)$$\\$ If the two vectors $\bar{a}$ and $\bar{b}$ act along a straight line in the same direction, then we can write: $\\$ $|\bar{a}+\bar{b}|=||\bar{a}|-|\bar{b}||\quad ...(v)$$\\ Combining equations (iv) and (v), we get: \\ |\bar{a}+\bar{b}| \geq ||\bar{a}|-|\bar{b}||$$\\$ Let two vectors a and b be represented by the adjacent sides of a parallelogram $PORS,$ as shown in the given figure.$\\$ $\\$