Motion In A Plane Class 11 Notes Physics Chapter 4 - CBSE
Chapter : 4
What Are Motion In A Plane ?
Scalars
The physical quantities which have only magnitude and no direction are called scalars e.g. length, time, mass, speed, work etc.
Vectors
The physical quantities which have both magnitude and direction are called vectors e.g., displacement, velocity, acceleration, force etc.
Zero Vector
A vector having zero magnitude and an arbitrary direction is called a zero or null vector.
Modulus Of A Vector
The magnitude or length of a vector is called its modulus.
$$\text{Modulus of vector}|\vec{\text{A}}| = \vec{\text{A}}= \text{A}$$
Unit Vector
A unit vector is a vector of unit magnitude drawn in the direction of a given vector.
$$\hat{\text{A}} =\frac{\vec{\text{A}}}{|\vec{\text{A}}|}$$
Composition Of Vectors
The resultant of two or more vectors is that single vector which produces the same effect as the individual vectors together would produced. The process of adding two ore more vectors is called the composition of vectors.
Triangle Law Of Vector Addition
It states that if two vectors can be represented both in magnitude and direction by the two sides of a triangle taken in the same order, then their resultant is represented completely both in magnitude and direction by the third side of the triangle taken in the opposite order.
Parallelogram Law Of Vector Addition
It states that if two vectors acting simultaneously at a point can be represented both in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented completely both in magnitude and direction by the diagonal of the parallelogram passing through that point.
$$\vec{\text{P}} + \vec{\text{Q}} = \vec{\text{R}}\\|\vec{R}| =\sqrt{\text{P}^{2} + \text{Q}^{2} +2\text{PQ cos}\theta}\\\text{tan}\space\beta =\frac{\text{Q sin}\space\theta}{\text{P + Q}\text{cos}\theta}$$
Rectangular Components Of A Vector
$$\vec{\text{A}} = \vec{\text{A}}_{x} + \vec{\text{A}}_{y}\\ = \text{A}_{X}\hat{i} + \text{A}_{y}\hat{j}$$
Ax = A cos θ
Ay = A sin θ
$$\text{A} =\sqrt{\text{A}^{2}_{x} + \text{A}^{2}_{y}}\\\text{tan}\space \theta =\frac{A_{y}}{A_{x}}$$
Scalar Or Dot Product
$$\vec{\text{A}}. \vec{\text{B}} =|\vec{\text{A}}||\vec{\text{B}}|\text{cos}\space \theta =\\\text{AB cos}\space\theta$$
Properties
$$\centerdot\space\text{For parallel vectors:}\\\vec{\text{A}}.\vec{\text{B}} = \text{AB}\\\centerdot\space\text{For perpendicular vectors :}\\\vec{\text{A}}.\vec{\text{B}} = 0$$
$$\centerdot\space\vec{\text{A}}.\vec{\text{B}} = \vec{\text{B}}.\vec{\text{A}}\\\text{(Commutative law)}\\\centerdot\space\vec{\text{A}}.(\vec{\text{B}} + \vec{\text{C}}) = \vec{\text{A}}.\vec{\text{B}} + \vec{\text{A}}.\vec{\text{C}}\\\text{(Distributive law)}\\\centerdot\space \vec{\text{A}}.\vec{\text{A}} = \text{A}^{2}\\\centerdot\space\hat{i}.\hat{i} = \hat{j}.\hat{j} = \hat{k}.\hat{k} = 1$$
Angle Between Two Vectors
$$\text{cos}\space\theta =\frac{\vec{\text{A}}.\vec{\text{B}}}{|A|.|B|}$$
Examples Of Dot Product
$$\centerdot\space\text{Work W} =\vec{F}.\vec{s}\\\centerdot\space \text{Power P =}\vec{\text{F}}.\vec{\text{v}}$$
Vector Product Or Cross Product
$$\vec{\text{A}}×\vec{\text{B}}=\\\text{AB sin}\space\theta.\hat{n}$$
$$\text{where}\space\hat{n}\space\text{is a unit vector perpendicular}\\\text{ to the plane of}\space\vec{\text{A}}\space\text{and}\space \vec{\text{B}}.$$
Properties
$$\centerdot\space\text{For parallel or antiparallel vectors,}\\\vec{\text{A}}×\vec{\text{B}} =0\\\centerdot\space\vec{\text{A}}×\vec{\text{B}} =-\vec{\text{B}}×\vec{\text{A}}\\\text{(Anti-commutative law)}\\\centerdot\space\vec{\text{A}}×(\vec{\text{B}}+ \vec{\text{C}}) =\\\vec{\text{A}}×\vec{\text{B}} +\vec{\text{A}}×\vec{\text{C}}\\\text{(Distributive law)}\\\centerdot\space \hat{i}×\hat{i} = \hat{j}×\hat{j} = \hat{k}×\hat{k} = 0\\\centerdot\space\hat{i}×\hat{j} = \hat{k}, \hat{j}×\hat{k}= \hat{i}, \hat{k}×\hat{i} = \hat{j}\\\centerdot\space\text{Angle between}\space\vec{A}\space\text{and}\space\vec{\text{B}}\\\text{sin}\space\theta =\frac{|\vec{\text{A}}×\vec{\text{B}}|}{|\vec{\text{A}}||\vec{\text{B}}|}n$$
Cross Product In Cartesian Coordinates
$$\text{A × B} =\begin{vmatrix}\hat{i} &\hat{j} &\hat{k}\\ A_{x} &A_{y} &A_{z}\\\text{B}_{x} &\text{B}_{y} &\text{B}_{z}\end{vmatrix} $$
$$=(A_{y}B_{z} - A_{Z}B_{y})\hat{i} +(\text{A}_{Z}\text{B}_{x} -\text{A}_{x}\text{B}_{z})\hat{j}+\\(\text{A}_{x}\text{B}_{y} - \text{A}_{y}\text{B}_{x})\hat{k}$$
Examples of cross product
$$\centerdot\space{\text{Torque}}\space\vec{\tau} =\vec{r}×\vec{\text{F}}\\\centerdot\space\text{Angular momentum,}\\\vec{\tau} =\vec{r}×\vec{p}$$
Projectile Motion
Any body projected into space such that it moves under the effect of gravity alone is called a projectile. The path followed by a projectile is called its trajectory, which is always a parabola.
Projectile Fired At An Angle With The Horizontal
$$\centerdot\space{\text{Equation of trajectory,}}\\ y = x\text{tan}\space\theta -\frac{gx^{2}}{2u^{2}\text{cos}^{2}\theta}\\\centerdot\space\text{Maximum height,}\\\text{H} =\frac{\text{U}^{2}\text{sin}^{2}\theta}{2g}\\\centerdot\space\text{Time of flight,}\\\text{T} =\frac{\text{2u sin}\space\theta}{g}\\\centerdot\space\text{Horizontal range,}\\\text{R} =\frac{u^{2}\text{sin}2\theta}{g}\\\centerdot\space\text{Maximum horizontal range,}\\\text{R} =\frac{v^{2}}{g}(at\space \theta = 45\degree)$$
Unifrom Circular Motion
When a body moves along a circular path with uniform speed, its motion is said to uniform circular motion.
Angular Displacement
It is the angle swept out by a radius vector in a given time interval.
$$\theta(\text{rad}) =\frac{\text{Arc}}{\text{radius}}=\frac{S}{r}$$
Angular Velocity
The angle swept out by the radius vector per second is called angular velocity.
$$\omega =\frac{\theta}{t}\text{or}\space\omega =\frac{\theta_{2}-\theta_{1}}{t_{2}-t_{1}}$$
Angular Acceleration
The rate of change of angular velocity is called angular acceleration:
$$\alpha =\frac{\omega_{2}-\omega_{2}}{t}$$
Also α = rα
i.e. linear acceleration = radius × angular acceleration
Time Period And Frequency
$$\text{T} =\frac{2\pi}{\omega}\space\text{and}\\\text{angular frequency}\space\omega = 2\pi n$$
Centripetal Acceleration
A body moving along a circular path is acted upon by an acceleration direction towards the centre along the radius. This acceleration is called centripetal acceleration.
$$a =\frac{v^{2}}{r} =r\omega^{2}$$
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