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HSSLIVE Plus One Physics Chapter 4: Motion in a Plane Notes

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Motion in a Plane expands students’ understanding beyond one dimension, exploring how objects move in two or three dimensions. This chapter introduces vector analysis for describing position, velocity, and acceleration in multiple dimensions. Students learn to solve projectile motion problems, analyze circular motion, and understand relative motion, gaining the tools to describe complex real-world movements like the path of a thrown ball or orbiting satellites.

Chapter 4: Motion in a Plane

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1. Introduction to Motion in a Plane

  • Definition: Movement of an object in two dimensions (x and y coordinates).
  • Examples: Projectile motion, circular motion, motion of planets, etc.
  • Types: Uniform and non-uniform motion in a plane.
  • Description: Requires vector quantities to fully describe the motion.

2. Scalars and Vectors

  • Scalar quantities: Quantities with magnitude only (e.g., mass, temperature, speed).
  • Vector quantities: Quantities with both magnitude and direction (e.g., displacement, velocity, acceleration, force).

2.1 Vector Representation

  • Notation: Vectors are represented by symbols with arrows above them (e.g., $\vec{A}$).
  • Graphical representation: Directed line segments with length proportional to magnitude.
  • Components: Vectors can be resolved into components along coordinate axes.

3. Position and Displacement Vectors

  • Position vector ($\vec{r}$): Vector from origin to the object’s location.
    • $\vec{r} = x\hat{i} + y\hat{j}$ (in 2D)
    • Magnitude: $|\vec{r}| = \sqrt{x^2 + y^2}$
  • Displacement vector ($\Delta\vec{r}$): Change in position vector.
    • $\Delta\vec{r} = \vec{r}_f – \vec{r}_i$
    • $\Delta\vec{r} = (x_f – x_i)\hat{i} + (y_f – y_i)\hat{j}$

4. Vector Addition and Subtraction

4.1 Vector Addition

  • Triangle Law: If two vectors are represented by two sides of a triangle taken in the same order, their resultant is represented by the third side taken in the reverse order.
  • Parallelogram Law: If two vectors are represented by two adjacent sides of a parallelogram, their resultant is represented by the diagonal passing through their common point.
  • Component Method: Add corresponding components.
    • $\vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$

4.2 Vector Subtraction

  • $\vec{A} – \vec{B} = \vec{A} + (-\vec{B})$
  • $\vec{A} – \vec{B} = (A_x – B_x)\hat{i} + (A_y – B_y)\hat{j}$

5. Multiplication of Vectors

5.1 Multiplication of a Vector by a Scalar

  • Changes the magnitude of the vector (and direction if scalar is negative).
  • $m\vec{A} = m(A_x\hat{i} + A_y\hat{j}) = mA_x\hat{i} + mA_y\hat{j}$

5.2 Scalar (Dot) Product

  • Definition: $\vec{A} \cdot \vec{B} = |A||B|\cos\theta$, where θ is the angle between vectors.
  • Component form: $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y$
  • Properties:
    • Commutative: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$
    • Distributive: $\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$
    • Result is a scalar quantity

5.3 Vector (Cross) Product

  • Definition: $\vec{A} \times \vec{B} = |A||B|\sin\theta, \hat{n}$, where θ is the angle between vectors and $\hat{n}$ is a unit vector perpendicular to both vectors.
  • Direction: Determined by right-hand rule.
  • Component form: $\vec{A} \times \vec{B} = (A_xB_y – A_yB_x)\hat{k}$ (in 2D)
  • Properties:
    • Anti-commutative: $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$
    • Distributive: $\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$
    • Result is a vector quantity

6. Velocity and Acceleration in a Plane

6.1 Velocity Vector

  • Average velocity: $\vec{v}_{avg} = \frac{\Delta\vec{r}}{\Delta t}$
  • Instantaneous velocity: $\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta\vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$
  • Components: $\vec{v} = v_x\hat{i} + v_y\hat{j}$, where $v_x = \frac{dx}{dt}$ and $v_y = \frac{dy}{dt}$
  • Magnitude: $|\vec{v}| = \sqrt{v_x^2 + v_y^2}$

6.2 Acceleration Vector

  • Average acceleration: $\vec{a}_{avg} = \frac{\Delta\vec{v}}{\Delta t}$
  • Instantaneous acceleration: $\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta\vec{v}}{\Delta t} = \frac{d\vec{v}}{dt}$
  • Components: $\vec{a} = a_x\hat{i} + a_y\hat{j}$, where $a_x = \frac{dv_x}{dt}$ and $a_y = \frac{dv_y}{dt}$
  • Magnitude: $|\vec{a}| = \sqrt{a_x^2 + a_y^2}$

7. Projectile Motion

7.1 Characteristics

  • Definition: Motion of an object thrown at an angle with the horizontal.
  • Path: Parabolic trajectory (neglecting air resistance).
  • Independence of components: Horizontal and vertical motions are independent.

7.2 Key Equations

  • Initial velocity: $\vec{u} = u\cos\theta,\hat{i} + u\sin\theta,\hat{j}$
  • Position: $\vec{r}(t) = (u\cos\theta)t,\hat{i} + [(u\sin\theta)t – \frac{1}{2}gt^2]\hat{j}$
  • Velocity: $\vec{v}(t) = (u\cos\theta)\hat{i} + [(u\sin\theta) – gt]\hat{j}$
  • Acceleration: $\vec{a} = -g\hat{j}$

7.3 Important Parameters

  • Time of flight: $T = \frac{2u\sin\theta}{g}$
  • Maximum height: $H = \frac{u^2\sin^2\theta}{2g}$
  • Horizontal range: $R = \frac{u^2\sin2\theta}{g}$
  • Equation of trajectory: $y = x\tan\theta – \frac{gx^2}{2u^2\cos^2\theta}$

7.4 Properties

  • Maximum range: Occurs at θ = 45°
  • Same range: Occurs for complementary angles (θ and 90° – θ)
  • Time of flight for horizontal projection: $T = \sqrt{\frac{2h}{g}}$, where h is the height of projection

8. Uniform Circular Motion

8.1 Characteristics

  • Definition: Motion of an object along a circular path with constant speed.
  • Speed: Magnitude of velocity remains constant.
  • Direction: Direction of velocity changes continuously (tangent to the circle).

8.2 Parameters

  • Angular displacement (θ): Angle swept by position vector.
  • Angular velocity (ω): Rate of change of angular displacement.
    • $\omega = \frac{d\theta}{dt}$
    • Unit: radian per second (rad/s)
  • Period (T): Time taken for one complete revolution.
    • $T = \frac{2\pi}{\omega}$
  • Frequency (f): Number of revolutions per unit time.
    • $f = \frac{1}{T} = \frac{\omega}{2\pi}$
    • Unit: hertz (Hz)

8.3 Relation Between Linear and Angular Quantities

  • Linear speed (v): $v = r\omega$
  • Linear displacement (s): $s = r\theta$

8.4 Centripetal Acceleration

  • Definition: Acceleration directed towards the center of the circular path.
  • Magnitude: $a_c = \frac{v^2}{r} = r\omega^2$
  • Direction: Radially inward (towards the center)
  • Cause: Centripetal force causing the circular motion.

9. Relative Motion in a Plane

9.1 Relative Velocity

  • Formula: $\vec{v}_{AB} = \vec{v}_A – \vec{v}_B$
    • $\vec{v}_{AB}$ = velocity of object A as observed from object B
    • $\vec{v}_A$ = velocity of object A in ground frame
    • $\vec{v}_B$ = velocity of object B in ground frame

9.2 Applications

  • River-boat problems: Finding effective velocity when moving in flowing water.
  • Wind-aircraft problems: Finding effective velocity when moving in wind.
  • Relative motion problems: Finding the apparent motion of one object as seen from another.

10. Important Points to Remember

  • Motion in a plane requires vector analysis.
  • Horizontal and vertical components of projectile motion are independent.
  • In projectile motion, horizontal velocity remains constant (neglecting air resistance).
  • In uniform circular motion, speed is constant but velocity changes due to changing direction.
  • Centripetal acceleration is always directed towards the center of the circular path.
  • Relative velocity is important when analyzing motion from different reference frames.

11. Summary

  • Motion in a plane involves movement in two dimensions, requiring vector descriptions.
  • Vectors can be added by component method, triangle law, or parallelogram law.
  • Projectile motion is a combination of uniform motion in horizontal direction and uniformly accelerated motion in vertical direction.
  • The path of a projectile is parabolic when air resistance is neglected.
  • Uniform circular motion involves constant speed but changing velocity direction.
  • Centripetal acceleration is necessary for maintaining circular motion.
  • Relative motion analysis helps understand how motion appears from different reference frames.

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