Quick Revision Notes: Class 11 Physics

Chapter: Motion in a Plane (समtल में गति)

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### 1. Scalars and Vectors (अदिश और सदिश राशियाँ)

• Scalar Quantity (अदिश राशि): Quantities having only magnitude and no direction (e.g., Mass, Distance, Time, Speed, Work).
• Vector Quantity (सदिश राशि): Quantities having both magnitude and direction, and obeying the triangle law of addition (e.g., Displacement, Velocity, Acceleration, Force).
• Position Vector ($\vec{r}$): A vector that specifies the position of an object with respect to the origin. $$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$$
• Unit Vector ($\hat{A}$): A vector of unit magnitude pointing in the direction of vector $\vec{A}$. $$\hat{A} = \frac{\vec{A}}{|\vec{A}|}$$

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### 2. Vector Addition and Subtraction (सदिशों का जोड़ और घटाव)

• Triangle Law of Addition: If two vectors are represented by two sides of a triangle in order, their sum (resultant) is given by the third side in the opposite order.
• Parallelogram Law of Addition: If two vectors $\vec{A}$ and $\vec{B}$ are adjacent sides of a parallelogram, their resultant $\vec{R}$ is given by the diagonal passing through the common tail.
  • Magnitude of Resultant ($R$): $$R = \sqrt{A^2 + B^2 + 2AB \cos\theta}$$
  • Direction ($\alpha$ with $\vec{A}$): $$\tan\alpha = \frac{B \sin\theta}{A + B \cos\theta}$$ (where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$)

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### 3. Resolution of Vectors (सदिशों का वियोजन)

A vector $\vec{A}$ in a 2D plane can be resolved into two rectangular components:

• $A_x = A \cos\theta$ (along x-axis)
• $A_y = A \sin\theta$ (along y-axis)
• Magnitude: $A = \sqrt{A_x^2 + A_y^2}$
• Direction: $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

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### 4. Product of Vectors (सदिशों का गुणनफल)

• Scalar Product / Dot Product (अदिश या बिंदु गुणन): $$\vec{A} \cdot \vec{B} = AB \cos\theta = A_x B_x + A_y B_y + A_z B_z$$
  • Condition for perpendicular vectors: $\vec{A} \cdot \vec{B} = 0 \implies \cos\theta = 0 \implies \theta = 90^\circ$
• Vector Product / Cross Product (सदिश या वज्र गुणन): $$\vec{A} \times \vec{B} = (AB \sin\theta)\hat{n}$$
  • Determinant form: $$\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix}$$
  • Condition for parallel vectors: $\vec{A} \times \vec{B} = 0 \implies \theta = 0^\circ \text{ or } 180^\circ$

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### 5. Projectile Motion (प्रक्षेप्य गति)

Motion of an object thrown obliquely in air, moving under the sole influence of gravity.

Let initial velocity be $u$ and angle of projection be $\theta$ with the horizontal.

• Equation of Path (Trajectory): $$y = x \tan\theta - \frac{g x^2}{2 u^2 \cos^2\theta}$$ (This is a parabola equation, $y = ax - bx^2$)

• Time of Flight ($T$): Total time for which the projectile remains in air. $$T = \frac{2u \sin\theta}{g}$$

• Maximum Height ($H$): Maximum vertical distance attained. $$H = \frac{u^2 \sin^2\theta}{2g}$$

• Horizontal Range ($R$): Total horizontal distance covered. $$R = \frac{u^2 \sin(2\theta)}{g}$$

  • Note: Range is maximum when $\theta = 45^\circ \implies R_{max} = \frac{u^2}{g}$

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### 6. Uniform Circular Motion (एकसमान वृत्तीय गति)

When an object moves in a circular path with constant speed.

• Angular Displacement ($\theta$): Angle swept by radius vector per unit time (measured in radians).
• Angular Velocity ($\omega$): Rate of change of angular displacement. $$\omega = \frac{d\theta}{dt} = 2\pi f = \frac{2\pi}{T}$$
• Relation between Linear and Angular Velocity ($v$ and $\omega$): $$v = r\omega$$
• Centripetal Acceleration ($a_c$): Acceleration directed towards the center of the circular path. $$a_c = \frac{v^2}{r} = r\omega^2$$
• Centripetal Force ($F_c$): $$F_c = \frac{m v^2}{r} = m r \omega^2$$

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### Key Points for MP Board Exams (महत्वपूर्ण बिंदु)

1. Always include proper vector arrows ($\vec{A}$) and unit vectors ($\hat{i}, \hat{j}, \hat{k}$) in vector derivations.
2. For maximum range in projectile motion, always mention $\theta = 45^\circ$.
3. Distinguish clearly between Centripetal Force (अभिमुखी बल) and Centrifugal Force.
4. Numerical problems frequently come from Projectile Motion formulas ($T$, $H$, $R$) and Vector dot/cross products.