NCERT Class 11 Physics Chapter 4 Motion in a Plane Notes - Download PDF

Have you ever watched a bird change direction mid-flight or seen a football curve through the air after a powerful kick? These real-life examples involve motion in a plane when an object moves in two dimensions. In These NCERT notes you will learn how to analyze and describe such motion using vectors and basic math tools.

NCERT Notes for Class 11 Physics Chapter 3: Motion in a Plane are very helpful if you are trying to score well in physics. In the last chapter, we looked at motion in a straight line using plus and minus signs to show direction. But in real life, things don’t just move in a straight line they move in all sorts of directions. That is where vectors come in. Vectors help us explain motion when both direction and size matter, like with speed, force, or displacement. To do well in exams, it is important to understand each topic properly and practice the NCERT questions at the end of the chapter. These NCERT Notes for class 11 are prepared by our expert faculty based on the latest CBSE syllabus.

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NCERT Notes for Class 11 Chapter 3

NCERT Notes on Class 11 Physics Chapter 3 Motion in a Plane gives a precise and concise description of two-dimensional motion, vectors, projectile motion and uniform circular motion. These notes can be used to revise the concepts instantly and prepare the objective exams such as CBSE, JEE, and NEET.

Scalars And Vectors

Scalar

• A scalar has only magnitude (size).

• You get it by multiplying the number with the unit of the quantity.

• Examples: Mass, Speed, Distance

• You can add, subtract, and multiply scalars using simple math.

Vector

• A vector has both magnitude and direction.

• They follow the rules of vector addition like: $\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{B}+\overrightarrow{A}$ and the law of parallelogram addition.

• Examples: Displacement, Velocity, Acceleration, Force
• The magnitude of a vector is represented by $|\overrightarrow{A}|$ or $A$. Vectors are shown with an arrow.

Representation of vector

• A vector looks like a line with an arrow.
• The length of the arrow shows the magnitude.
• The direction of the arrow shows the direction of the vector.

A vector can be represented geometrically as a directed line segment with an arrowhead. The arrow's length represents the magnitude of the vector, and it points in the same direction as the vector itself.Tail $\underset{\text{ (magnitude) }}{\stackrel{\text{ Length }}{\to }}$ Head

Types of Vector

• Equal vectors: If the magnitudes and directions of vectors A and B are the same, they are equal.
• Parallel vectors are A and B when: 1. They both point in the same direction. 2. Two vectors can be expressed as scalar (positive) non-zero multiples of each other.
• Antiparallel vectors: A and B are anti-parallel if their directions are opposite.
  One vector is a negative multiple of the other that is not zero.
  A vector with 0 magnitude and an arbitrary (unknown) direction is called a zero vector.

• Unit vector: A unit vector is a vector with a fixed magnitude that points in a specific direction. A vector (A) can be expressed as the product of a unit vector (Â) in its direction and magnitude.

$\overrightarrow{A}=A\hat{A}$ or $\hat{A}=\frac{\overrightarrow{A}}{A}$

• A unit vector has no dimensions or units. Unit vectors along the positive x, y, and z axes of a rectangular coordinate system are denoted by î, ˆ j, and k̂, respectively. such that $|\hat{\mathbf{i}}|=|\hat{\mathbf{j}}|=|\hat{\mathbf{k}}|=1$

Position and Displacement Vectors

A position vector is a vector that represents the location of a point or a particle in space with respect to a fixed origin. It is usually denoted by $\overrightarrow{r}$ and is drawn from the origin to the location of the object.

If a particle is located at point $P(x,y,z)$, then its position vector is:

$\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$

A displacement vector represents the change in position of a particle. It is defined as the vector that points from the initial position to the final position of the object.

If the position vector of the initial point is ${\overrightarrow{r}}_1$ and the final point is ${\overrightarrow{r}}_2$, then the displacement is given by:

$\overrightarrow{\mathrm{\Delta }r}={\overrightarrow{r}}_2-{\overrightarrow{r}}_1$
Displacement is a vector quantity, having both magnitude and direction, and is independent of the actual path taken.

Equality of Vectors

Two vectors are said to be equal if they have the same magnitude and same direction, regardless of their initial points.

That is, if $\overrightarrow{A}$ and $\overrightarrow{B}$ satisfy:

$|\overrightarrow{A}|=|\overrightarrow{B}|\text{ and }\text{ Direction of }\overrightarrow{A}=\text{ Direction of }\overrightarrow{B}$

then $\overrightarrow{A}=\overrightarrow{B}$.
This means that vectors can be moved parallel to themselves in space without changing their properties, as long as magnitude and direction are preserved.

Multiplication of Vectors by Real Numbers

If $\overrightarrow{A}$ is a vector and $\lambda$ is a real number (scalar), then the product $\lambda \overrightarrow{A}$ is also a vector.

• The magnitude of $\lambda \overrightarrow{A}$ is $|\lambda ||\overrightarrow{A}|$.
• The direction of $\lambda \overrightarrow{A}$ is the same as $\overrightarrow{A}$ if $\lambda >0$, and opposite to $\overrightarrow{A}$ if $\lambda <0$.

Addition and Subtraction of Vectors — Graphical Method

Either the triangle law or the parallelogram law can be used to add vectors:

(A) Parallelogram law of addition of vectors: The diagonal drawn through the intersection of two vectors, A and B, represents the resultant vector if they are represented by two adjacent sides of a parallelogram, both pointing outward from a common point (with their tails coinciding).

$R=\sqrt{P^2+Q^2+2PQ\cos \theta }$

(B) The Triangle Law of Vector Addition states that if two vectors are represented by two triangle sides, then the third side of the triangle represents their total or resultant vector, but in the opposite direction.

Subtraction of Vectors

To subtract vector $\overrightarrow{B}$ from $\overrightarrow{A}$, we add $\overrightarrow{A}$ and the negative of vector $\overrightarrow{B}$.

$\overrightarrow{A}-\overrightarrow{B}=\overrightarrow{A}+(-\overrightarrow{B})$

• Reverse the direction of $\overrightarrow{B}$ to get $-\overrightarrow{B}$.
• Then use either the triangle law or parallelogram law to add $\overrightarrow{A}$ and $-\overrightarrow{B}$.

Multiplication of Vectors

Scalar Product (Dot Product): The scalar product or dot product of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, denoted as $\overrightarrow{a}\cdot \overrightarrow{b}$ , is defined as the product of their magnitudes multiplied by the cosine of the angle (θ) between them. Mathematically, it is expressed as:

$\overrightarrow{a}\cdot \overrightarrow{b}=a_x{b}_x+a_y{b}_y+a_z{b}_z=ab\cos \theta$

Vector Product (Cross Product): The vector product or cross product of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, denoted as $\overrightarrow{a}X\overrightarrow{b}$ B, is a vector quantity defined as follows:

$\begin{array}{c}\overrightarrow{a}\times \overrightarrow{b}=(a_y{b}_z-a_z{b}_y)\hat{ı}+(a_z{b}_x-a_x{b}_z)\hat{ȷ}+(a_x{b}_y-a_y{b}_x)\hat{k}\\ |\overrightarrow{a}\times \overrightarrow{b}|=ab\sin \theta \end{array}$

Lami’s Theorem:

• Lami's Theorem states that each force is proportional to the sine of the angle generated by the other two forces if three forces operating at the same spot are in equilibrium.

$\frac{F_1}{\sin \alpha }=\frac{F_2}{\sin \beta }=\frac{F_3}{\sin \gamma }$

Resolution of Vectors

Resolution of a vector is the process of expressing a single vector as the sum of two or more vectors (called components), usually along mutually perpendicular directions (like the $x$-axis and $y$-axis).
If a vector $\overrightarrow{A}$ makes an angle $\theta$ with the x-axis, then:

$\begin{array}{ll}{\overrightarrow{A}}_x=A\cos \theta & \text{ (component along }\mathrm{x}\text{-axis) }\\ {\overrightarrow{A}}_y=A\sin \theta & \text{ (component along }\mathrm{y}\text{-axis) }\end{array}$

Thus, the vector $\overrightarrow{A}$ can be written as:

$\overrightarrow{A}={\overrightarrow{A}}_x\hat{i}+{\overrightarrow{A}}_y\hat{j}=A\cos \theta \hat{i}+A\sin \theta \hat{j}$

Vector Addition – Analytical Method

Let two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ be added. Suppose:
$\overrightarrow{A}$ has components $A_x$ and $A_y$, and
$\overrightarrow{B}$ has components $B_x$ and $B_y$

Then the resultant vector $\overrightarrow{R}=\overrightarrow{A}+\overrightarrow{B}$ has components:

$\begin{array}{rl}& R_x=A_x+B_x\\ & R_y=A_y+B_y\end{array}$

Now, the magnitude of the resultant vector is:

$|\overrightarrow{R}|=\sqrt{R_x^2+R_y^2}=\sqrt{{(A_x+B_x)}^2+{(A_y+B_y)}^2}$

And the direction (angle $\theta$ with the $x$-axis) is given by:

$\tan \theta =\frac{R_y}{R_x}=\frac{A_y+B_y}{A_x+B_x}$

Motion in a Plane

In motion in a plane, the position, velocity, and acceleration of an object are all described using vectors.

The motion can be analyzed by breaking it into two perpendicular directions, usually the $\mathbf{x}$ - and $\mathbf{y}$ axes.

The position vector of an object at any instant is given by:

$\overrightarrow{r}=x(t)\hat{i}+y(t)\hat{j}$

The velocity vector is the time derivative of the position vector:

$\overrightarrow{v}=\frac{d\overrightarrow{r}}{dt}=v_x\hat{i}+v_y\hat{j}$

The acceleration vector is the derivative of velocity:

$\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}=a_x\hat{i}+a_y\hat{j}$

Projectile Motion

A body that is propelled with some initial velocity—not including vertical upward or downward motion—is called a projectile. Once in motion, the projectile moves only due to gravity; it is not further propelled by an engine, fuel, or other external source. A projectile's trajectory is the course it takes while in motion.

• For motion along the X-axis,

$v_x=u_x+a_{x}t\text{ and }x=x_0+u_{x}t+\frac{1}{2}{a}_x{t}^2$

- For motion along $\underset{―}{\mathrm{Y}\text{-axis, }}$

$v_y=u_y+a_{y}t\text{ and }y=y_0+u_{y}t+\frac{1}{2}{a}_y{t}^2$

• Angular projection of projectile :

1. Time of flight ( T ):

$\mathrm{T}=\frac{2u\sin \theta }{g}$

2. Maximum height(h):

$h=\frac{u^2{\sin }^2\theta }{2g}$

3. Horizontal range(R):

$\mathrm{R}=\frac{u^2\sin 2\theta }{g}$

4. Maximum horizontal range( ${\mathrm{R}}_{max}$ ):

$R_{max}=\frac{u^2}{g}\text{ for }\theta ={45}^{\circ }$

Note: For maximum range, θ should be 45 degrees.

Equation of trajectory

A trajectory is the term used to describe the body's journey. We must determine the link between y and x and eliminate time in order to build the trajectory.

Uniform Circular Motion

When an object moves in a circular path with constant speed, the motion is called Uniform Circular Motion (UCM). Although the speed remains constant, the direction of velocity changes continuously, making it an accelerated motion.

• Speed is constant, but velocity changes due to change in direction.
• The object experiences a centripetal acceleration directed towards the center of the circular path.
• A centripetal force is required to keep the object moving in the circle, also directed towards the center.

• Angular Velocity:
  $\omega =\frac{2\pi }{T}(\text{ radians per second })$

• Relation between $v$ and $\omega$ :
  $v=r\omega$

• Centripetal Acceleration:
  $a_c=\frac{v^2}{r}=r{\omega }^2$

• Centripetal Force: $F_c=m{a}_c=\frac{m{v}^2}{r}$

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