RBSE Class 11 Physics Notes Chapter 4 Motion in a Plane

These comprehensive RBSE Class 11 Physics Notes Chapter 4 Motion in a Plane will give a brief overview of all the concepts.

RBSE Class 11 Physics Chapter 4 Notes Motion in a Plane

Scalar quantities:
The quantities which require only magnitude for their representation, are called scalar quantities. For example: distance, volume, speed etc.

Vector quantities:
The quantities which require magnitude as well as direction both for their representation, are called vector quantities. These quantities obey the vector laws of addition and subtraction. For example: displacement, velocity, acceleration etc.

Kinds of vectors:

• Axial vectors: These are the vectors which show rotational effect and their directions are decided by right hand screw rule, are called axial vectors. For example: angular velocity (ω), angular acceleration (α) etc.
• Polar vectors: These are the vectors which have certain initial point, are called polar vectors. For example: displacement, force etc.

 

Addition of vectors:

• Triangle law: According to this law, if two vectors can be represented in magnitude and direction both by two arms of a triangle, then the magnitude of their resultant will be represented by third arm of the triangle.
• Law of parallelogram: If two vectors can be represented by two adjoining arms of parallelogram in magnitude and direction both, then their resultant will be represented in magnitude and direction both by that diagonal of the parallelogram which passes through the intersection point of those adjoining arms.
• Law of polygon: If (n -1) vectors can be represented in magnitude and direction both by (n - 1) arms of a polygon of n arms, then magnitude and direction of resultant vector will be represented by last arm of the polygon in opposite direction.

Subtraction of vector:
Process of subtraction is also the process of addition, the vector which is to be subtracted is added with its negative value.

Position vector:
The vector which represents the position of a point, is called the position vector of that point.

Displacement vector:
The vector which represertt the displacement of a moving object, is called displacement vector.

Relative velocity:
The velocity of an object with respect to other object is called the relative velocity of first body with respect to other body. For example: relative velocity of A with respect to B
V_AB = V_A - V_B

Product of vectors:
(i) Product of scalar by vector: When a vector is multiplied by a scalar, then product is a vector quantity. The magnitude of obtained vector is the product of magnitudes of both quantities and its direction is same as that given vector. For Example: force \(\vec{F}=m \vec{a}\) and linear momentum \(\vec{p}=m \vec{v}\)

(ii) Scalar product or dot product: This product of two vectors is a scalar quantity, therefore it is called scalar product and it is denoted by symbol of dot (.), therefore it is called dot product also. This product of two vectors is given by:
\(\vec{A} \cdot \vec{B}=|\vec{A}||\vec{B}|\)cos θ
or \(\vec{A} \cdot \vec{B}\) = ABcos θ

(iii) Vector product or cross-product: This product of two vectors is a vector quantity, therefore it is called vector product and it is denoted by symbol of cross (x), therefore it is also called cross prodcut.
\(\vec{A} \times \vec{B}=|\vec{A}| .|\vec{B}| \sin \theta \hat{n}\) or \(\vec{A} \times \vec{B}=A \cdot B \sin \theta \cdot \hat{n}\)
Where n̂ is unit vector at right angle to plane of \(\vec{A}\) and \(\vec{B}\), direction of which is derived by right handed screw rule.

Two dimensional motion:
When two co-ordinates are required to represent the motion of a particle, then it is called two dimensional motion.

Projectile motion:
The object which move on parabolic path, is called projectile and its path is called trajectory.

Uniform circular motion:
When an object moves on a circular path with uniform speed then it is called uniform circular motion.

• Radius vector: The line joining the centre of the circular path and the object is called radius vector.
• Angular position: The angle made by radius vector at any time with its initial position, is called the angular position of the object at that time.
• Angular displacement: The difference of angular position with respect to certain time interval is called its angular displacement
  i. e. Δθ = (θ₂ - θ₁).
• Angular velocity: The rate of change of angular displacement with time, is called angular velocity. It is denoted by to and its unit is rad s^-1
  ω = \(\frac{\Delta \theta}{\Delta t}=\frac{d \theta}{d t}\)
• Angular acceleration: Rate of angular velocity with time is called angular acceleration and it is denoted by a.
  α = \(\frac{\Delta \omega}{\Delta t}=\frac{d \omega}{d t}\)
• Relation between ω ,r and v = rω
• Relation between a,r and α i.e a = r. α
• The resultant acceleration in circular motion is \(\sqrt{a_R^2+a_T^2}\)

Centripetal acceleration:
In uniform circular motion an acceleration is active on the particle, the direction of which remains always toward the centre, it is called centripetal acceleration.

→ Unit vector in direction of vector B,B̂ = \(\frac{\vec{B}}{|\vec{B}|}\)

→ Vector addition obeys association law i.e.,
\((\vec{A}+\vec{B})+\vec{C}=\vec{A}+(\vec{B}+\vec{C})\)

→ Vector addition is commulative i. e.
\(\vec{A}+\vec{B}=\vec{B}+\vec{A}\)

→ Vector addition is distributive i.e.
(m+n) \(\vec{A}\) = m \(\vec{A}\) + n \(\vec{A}\)

→ Magnitude of resultant of two vectors by triangle or parallelogram law is obtained as
R = \(\sqrt{P^2+Q^2+2 P Q \cos \alpha}\)
where θ is angle between \(\vec{P}\) and \(\vec{Q}\)
If resultant \(\vec{R}\) makes an angle θ with \(\vec{P}\), then
α = tan^-1\(\left[\frac{Q \cdot \sin \theta}{P+Q \cos \theta}\right]\)

→ If a vector \(\vec{A}\) makes an angle θ with X-axis, then its components,
A_x = A cos θ and A_y = A sin θ

→ If position vector of A₁ and A₂ are \(\vec{r}_1 \)and \(\vec{r}_2\) respectively, then displacement vector,
A₁A₂ = \(\overrightarrow{r_2}-\overrightarrow{r_1}\)

→ If a vector \(\vec{A}\) makes angles α, β, γ with X,Y and Z axes respectively, then
\(|\vec{A}|=\sqrt{A_x^2+A_y^2+A_z^2}\)
and cos²α + cos²β + cos²γ = 1
and sin²α + sin²β + sin²γ = 2

→ If velocities of moving bodies 1 and 2 are respectively \(\overrightarrow{v_1}\) and \(\overrightarrow{v_2}\), then relative velocity of 2 with respect to 1,
\(\overrightarrow{v_{21}}=\overrightarrow{v_2}-\overrightarrow{v_1}\)

→ If a swimmer can swim in still water with velocity \(\overrightarrow{v_m}\) and velocity of river flow be \(\overrightarrow{v_r}\), then
(i) Time taken in crossing the river in minimum distance,
t = \(\frac{d}{\sqrt{v_m^2-v_r^2}}\) where d is width of the river.

(ii) Time taken in minimum time
t = \(\frac{d}{v_m}\)

→ Scalar product of \(\vec{A}\) and \(\vec{B}\),
\(\vec{A} \cdot \vec{B}=|\vec{A}| \cdot|\vec{B}|\) cos θ = ABcosθ
(i) Scalar product is commutative, i. e.,
\(\vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{A}\)
(ii) Scalar product of a vector with itself
\(\vec{A} \cdot \vec{A}\) = Acos θ°= A A₁ = A²
(iii) Dot product of orthogonal unit vectors,
î.î = ĵ.ĵ = k̂k̂= 1
î.ĵ = ĵ.î = î.k̂ = k̂.î = ĵ.k̂ = k̂.ĵ = 0

→ Multiplication of a vector by scalar,
\(\vec{A}=m \vec{a} and \vec{p}=m \vec{v}\)

→ Cross product of two vectors,
\(\vec{A} \times \vec{B}=|\vec{A}| \cdot|\vec{B}|\)sin θn̂ = A.B sin θ n̂
where n̂ = unit vector in direction perpendicular to plane of \(\vec{A}\) and \(\vec{B}\), decided by right hand screw law:
(i) Cross-product is not commulative,
\(\vec{A} \times \vec{B}=-(\vec{B} \times \vec{A})\)

(ii) Vector product of a vector with itself
\(\vec{A} \times \vec{A}\) = A Asin0°.n̂ = 0

(iii) Cross product of orthogonal unit vectors
î × î = ĵ × ĵ = k̂ × k̂ = 0
î × ĵ = k̂; ĵ × î = -k̂
î × k̂ = -ĵ; k̂ × î = ĵ
ĵ × k̂ = î; k̂ × ĵ = -î

→ If (x, y) be the co-ordinates of a point, then its position vector,
\(\vec{v}\) = xî + y ĵ

→ If a moving particle is at point P(x₁, y₁) at time and at a(x₂, y₂) at time t₂, then position vectors of P_E and Q are
\(\overrightarrow{r_1} \) = x₁î + y₁ĵ and \(\overrightarrow{r_2}\) = x₂ î + y₂ ĵ
And displacement \(\overrightarrow{P Q}+\overrightarrow{\Delta r}=\overrightarrow{r_2}-\overrightarrow{r_1}\)
or \(\overrightarrow{\Delta r}\) = Δxî + Δy ĵ where Δx = x₂ - x₁ and Δy = y₂ - y₁

→ Velocity \(\vec{v}=\frac{\overrightarrow{\Delta r}}{\Delta t}=\frac{\Delta x \hat{i}+\Delta y \hat{j}}{\Delta t}=\frac{\Delta x}{\Delta t} \hat{i}+\frac{\Delta y}{\Delta t} \hat{j}\)
or \(\vec{v}\) = v_xî + v_yĵ
Magnitude of this velocity
\(|\vec{v}|=v=\sqrt{v_x^2+v_y^2}\)

→ Acceleration
\(\vec{a}=\frac{\overrightarrow{\Delta v}}{\Delta t}=\frac{v_x}{\Delta t} \cdot \hat{i}+\frac{v_y}{\Delta t}, \hat{j}=a_x \hat{i}+a_y \hat{j}\)
and \(|\vec{a}|=\sqrt{a_x^2+a_y^2}\)

→ If a projectile is projected with velocity u in horizontal direction from height h, then
(a) Time of flight T = \(\sqrt{\frac{2 h}{g}}\)
(ii) Horizontal range R = u × T

→ If projectile is projected with velocity u making an angle with horizontal direction, then

• Time of flight, T = \(\frac{2 u \sin \theta}{g}\)
• Horizontal range, R = \(\frac{u^2 \sin 2 \theta}{g}\)
• Maximum height, H = \(\frac{u^2 \sin ^2 \theta}{2 g}\)
• Equation of trajectory y = xtan θ - \(\frac{1}{2} \frac{g x^2}{u^2 \cos ^2 \theta}\)
• If a projectile is at same height (y) at time t₁ and t₂ then y = \(\frac{1}{2}\)gt₁t₂
• Angular displacement Δθ = (θ₂ - θ₁)
• Angular velocity ω = \(\frac{\Delta \theta}{\Delta t}=\frac{d \theta}{d t}\)
• Angular acceleration α = \(\frac{\Delta \omega}{\Delta t}=\frac{d \omega}{d t}\)
• Linear velocity v = \(\frac{\Delta S}{\Delta t}=\frac{d S}{d t}\)
• Linear acceleration a = \(\frac{\Delta v}{\Delta t}=\frac{d v}{d t}\)
• Relation between v and ω, v = rω
• Relation between a and a, α = rα
• Centripetal acceleration
  a_c = \(\frac{v^2}{r}\) = rω² = r.\(\frac{4 \pi^2}{T^2}\) = r.4π²n²
• If there is tangential acceleration also along with centripetal acceleration, the resultant acceleration.
  a = \(\sqrt{a_C^2+a_T^2}\)

→ Scalar quantities: The quantities which require only magnitude for their representation, are called scalar quantities.

→ Vector quantities: The quantities which require magnitude and direction both for their representation, are called vector quantities.

→ Tensor quantities: The quantities having magnitude and direction both, can not be defined completely, are called tensor quantities. For example: moment of inertia, modulus of elstity.

→ Axial vectors: The vectors which exhibit rotational effect, are called axial vectors.

→ Polar vectors: The vectors which have certain initial point, are called polar vectors.

→ Projectile: The object which moves on parabolic path, is called projectile.

→ Trajectory: The path of projectile is called trajectory.

→ Time of flight: The time for which projectile remains in air is called the time of flight.

→ Radius vector: The radius which decides the position of the particle in circular motion, is called radius vector.

→ Centripetal acceleration: In circular motion, an acceleration directing towards centre of the path always acts on the particle. This acceleration is called centripetal acceleration.