Class 11 Physics Notes Chapter 4 (Motion in a plane) – Physics Part-I Book

Alright class, let's delve into Chapter 4: Motion in a Plane. This is a crucial chapter as it builds upon your understanding of motion in a straight line and introduces the concept of vectors, which are fundamental to describing motion in two or three dimensions. This is essential not just for your Class 11 exams but also forms the bedrock for many topics in competitive government exams.

Chapter 4: Motion in a Plane - Detailed Notes

1. Introduction: Why Vectors?

• Motion in a straight line (Chapter 3) could be described using positive and negative signs to indicate direction.
• However, when an object moves in a plane (2D) or space (3D), we need a more sophisticated way to represent direction along with magnitude. This is where vectors come in.
• Examples of motion in a plane: Projectile motion (a ball thrown), Circular motion (Earth around the Sun), a car turning on a curved road.

2. Scalars and Vectors

• Scalar Quantity: A physical quantity that has only magnitude and no direction. It is specified by a single number, along with the proper unit.
  • Examples: Distance, speed, mass, temperature, time, work, energy, density.
  • Scalars are added, subtracted, multiplied, or divided using ordinary rules of algebra.
• Vector Quantity: A physical quantity that has both magnitude and direction. It must also obey the laws of vector addition (like the Triangle Law or Parallelogram Law).
  • Examples: Displacement, velocity, acceleration, force, momentum, torque.
  • Notation: Represented by an arrow over the symbol (e.g., v or \vec{v}) or by boldface type (v). The length of the arrow is proportional to the magnitude, and the arrowhead indicates the direction.
  • Magnitude of a vector A is denoted by |A| or simply A.

3. Vector Operations

• Equality of Vectors: Two vectors A and B are equal if they have the same magnitude and the same direction, regardless of their initial positions.
• Addition of Vectors:
  • Triangle Law: If two vectors are represented in magnitude and direction by the two sides of a triangle taken in order, then their sum (resultant) is represented in magnitude and direction by the third side of the triangle taken in the opposite order.
  • Parallelogram Law: If two vectors are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a common point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that common point.
  • If R = A + B, and θ is the angle between A and B, the magnitude of the resultant is:
    R = |**R**| = √(A² + B² + 2AB cos θ)
  • The direction of the resultant R (angle α with vector A) is given by:
    tan α = (B sin θ) / (A + B cos θ)
  • Properties:
    • Vector addition is commutative: A + B = B + A
    • Vector addition is associative: (A + B) + C = A + (B + C)
• Subtraction of Vectors: Subtracting vector B from A is the same as adding vector -B (a vector with the same magnitude as B but opposite direction) to A.
  A - B = A + (-B)
  Magnitude: |**A** - **B**| = √(A² + B² - 2AB cos θ) (where θ is the angle between A and B)
• Multiplication of a Vector by a Scalar: Multiplying a vector A by a scalar λ results in a vector λA.
  • Magnitude: |λA| = |λ| |A|
  • Direction: Same as A if λ > 0, opposite to A if λ < 0.
• Null Vector (Zero Vector): A vector with zero magnitude and an arbitrary direction. Denoted by 0.
  • Properties: A + 0 = A; λ0 = 0; 0A = 0.
• Unit Vector: A vector with a magnitude of 1 (unity). It is used to specify a direction.
  • A unit vector in the direction of vector A is denoted by  (read as 'A cap' or 'A hat').
  •  = **A** / |**A**| or **A** = |**A**| Â
  • Orthogonal Unit Vectors: In a right-handed Cartesian coordinate system (x, y, z), the unit vectors along the positive x, y, and z axes are denoted by î, ĵ, and k̂, respectively. They are mutually perpendicular. |î| = |ĵ| = |k̂| = 1.

4. Resolution of Vectors

• Any vector A in a plane can be split into two component vectors along two specified directions. It's most useful to resolve into perpendicular components (rectangular components).
• If vector A makes an angle θ with the x-axis:
  • x-component: Ax = A cos θ
  • y-component: Ay = A sin θ
• In vector form: **A** = Ax **î** + Ay **ĵ**
• Magnitude from components: A = |**A**| = √(Ax² + Ay²)
• Direction from components: tan θ = Ay / Ax
• In 3D: **A** = Ax **î** + Ay **ĵ** + Az **k̂
A = |**A**| = √(Ax² + Ay² + Az²)

5. Vector Addition using Components (Analytical Method)

• This is often the easiest way to add multiple vectors.
• Resolve each vector into its x and y components.
• Add all x-components algebraically to get the x-component of the resultant (Rx).
• Add all y-components algebraically to get the y-component of the resultant (Ry).
  **R** = **A** + **B**
Rx = Ax + Bx
Ry = Ay + By
**R** = Rx **î** + Ry **ĵ**
• Magnitude of Resultant: R = √(Rx² + Ry²)
• Direction of Resultant: tan α = Ry / Rx (where α is the angle R makes with the x-axis).

6. Motion in a Plane: Kinematic Quantities as Vectors

• Position Vector (r): A vector drawn from the origin of the coordinate system to the position of the object.
  **r** = x **î** + y **ĵ** (in 2D)
• Displacement Vector (Δr): The change in position vector. If an object moves from position r₁ to r₂, the displacement is:
  Δ**r** = **r₂** - **r₁** = (x₂ - x₁) **î** + (y₂ - y₁) **ĵ** = Δx **î** + Δy **ĵ**
  • Displacement is the straight-line vector from the initial to the final point. Its magnitude is not necessarily the distance travelled.
• Velocity Vector (v):
  • Average Velocity: **v**_avg_ = Δ**r** / Δt (Direction is the same as Δr)
  • Instantaneous Velocity: The limit of the average velocity as the time interval approaches zero. It is the time rate of change of position.
    **v** = lim (Δt→0) [Δ**r** / Δt] = d**r** / dt
  • Direction of v is tangent to the path at that point.
  • Components: **v** = vx **î** + vy **ĵ where vx = dx/dt and vy = dy/dt
  • Magnitude (Speed): v = |**v**| = √(vx² + vy²)
• Acceleration Vector (a):
  • Average Acceleration: **a**_avg_ = Δ**v** / Δt = (**v₂** - **v₁**) / (t₂ - t₁) (Direction is the same as Δv)
  • Instantaneous Acceleration: The limit of the average acceleration as the time interval approaches zero. It is the time rate of change of velocity.
    **a** = lim (Δt→0) [Δ**v** / Δt] = d**v** / dt
  • Components: **a** = ax **î** + ay **ĵ where ax = dvx/dt = d²x/dt² and ay = dvy/dt = d²y/dt²
  • Acceleration can result from a change in speed, a change in direction, or both.

7. Motion in a Plane with Constant Acceleration

• If acceleration a is constant (both magnitude and direction), we can generalize the 1D kinematic equations.
• Let v₀ be the initial velocity at t=0 and v be the velocity at time t.
• Let r₀ be the initial position at t=0 and r be the position at time t.
• Equations:
  1. **v** = **v₀** + **a** t
  2. **r** = **r₀** + **v₀** t + (1/2) **a** t²
• These vector equations can be broken down into component equations:
  • Along x-axis:
    vx = v₀x + ax t
x = x₀ + v₀x t + (1/2) ax t²
  • Along y-axis:
    vy = v₀y + ay t
y = y₀ + v₀y t + (1/2) ay t²
• Motion in a plane with constant acceleration can be treated as two independent, simultaneous motions along perpendicular axes.

8. Relative Velocity in Two Dimensions

• The velocity of object A relative to object B is: **v**_AB_ = **v**_A_ - **v**_B_
• Similarly, the velocity of object B relative to object A is: **v**_BA_ = **v**_B_ - **v**_A_ = - **v**_AB_
• This concept is crucial for problems like:
  • Rain falling vertically, but appearing slanted to a moving observer (Rain-Man problems).
  • A boat crossing a river with flowing water (Boat-River problems). Determine resultant velocity and time to cross.

9. Projectile Motion

• An object thrown into space, moving under the influence of gravity alone (neglecting air resistance).
• It's an example of motion in a plane with constant acceleration (a = -g ĵ, if y-axis is vertically upwards, x-axis horizontal).
• Assumptions: Air resistance is negligible, g is constant, Earth's curvature is ignored.
• Initial velocity v₀ makes an angle θ₀ with the horizontal.
  • Initial horizontal velocity: v₀x = v₀ cos θ₀ (remains constant as ax = 0)
  • Initial vertical velocity: v₀y = v₀ sin θ₀ (changes due to gravity, ay = -g)
• Key Formulas:
  • Equation of Trajectory: (Path followed)
    y = (tan θ₀) x - (g / (2 (v₀ cos θ₀)²)) x²
    This is an equation of a parabola (y = ax + bx²), proving the path is parabolic.
  • Time of Flight (T): Total time the projectile is in the air.
    T = (2 v₀ sin θ₀) / g
  • Maximum Height (H): Highest vertical position reached.
    H = (v₀² sin² θ₀) / (2g)
  • Horizontal Range (R): Total horizontal distance covered.
    R = (v₀² sin(2θ₀)) / g
  • Maximum Range: Occurs when sin(2θ₀) is maximum (i.e., 1), which means 2θ₀ = 90°, or θ₀ = 45°.
    R_max = v₀² / g
  • Note: For a given v₀, the range is the same for angles θ₀ and (90° - θ₀).

10. Uniform Circular Motion (UCM)

• Motion of an object in a circle at a constant speed.
• Although the speed is constant, the velocity is continuously changing because the direction is changing.
• Therefore, there must be an acceleration. This acceleration is directed towards the center of the circle and is called Centripetal Acceleration (a_c).
• Magnitude of Centripetal Acceleration:
  a_c = v² / r (where v is the constant speed, r is the radius of the circle)
• Angular Velocity (ω): Rate of change of angular position (angle swept per unit time). Unit: rad/s.
  ω = Δθ / Δt (for uniform motion)
• Relation between linear speed (v) and angular velocity (ω):
  v = ω r
• Centripetal acceleration in terms of ω:
  a_c = (ωr)² / r = ω² r
• Time Period (T): Time taken to complete one full revolution.
  T = 2πr / v = 2π / ω
• Frequency (ν or f): Number of revolutions completed per unit time. Unit: Hertz (Hz) or s⁻¹.
  ν = 1 / T = ω / 2π
• The force providing the centripetal acceleration is called the Centripetal Force (e.g., tension in a string, gravitational force, friction). It is not a new kind of force, but rather the role played by an existing force. F_c = m a_c = mv² / r = mω²r.

Multiple Choice Questions (MCQs)

1. Which of the following is a vector quantity?
   a) Speed
   b) Mass
   c) Work
   d) Acceleration

2. A particle moves from position r₁ = (2î + 3ĵ) m to r₂ = (4î + 1ĵ) m in 2 seconds. The average velocity of the particle is:
   a) (î - ĵ) m/s
   b) (2î - 2ĵ) m/s
   c) (î + ĵ) m/s
   d) (3î + 2ĵ) m/s

3. The horizontal range of a projectile fired at an angle of 15° is 50 m. If it is fired with the same speed at an angle of 45°, its range will be:
   a) 60 m
   b) 71 m
   c) 100 m
   d) 141 m

4. For an object undergoing Uniform Circular Motion, which quantity remains constant?
   a) Velocity
   b) Acceleration
   c) Speed
   d) Displacement

5. The angle between vectors A = î + ĵ and B = î - ĵ is:
   a) 0°
   b) 45°
   c) 90°
   d) 180°

6. A boat is sent across a river with a velocity of 8 km/h. If the resultant velocity of the boat is 10 km/h, the velocity of the river flow is:
   a) 12.8 km/h
   b) 6 km/h
   c) 8 km/h
   d) 10 km/h

7. At the highest point of its trajectory, a projectile has:
   a) Maximum speed
   b) Minimum speed (but not zero if θ₀ ≠ 90°)
   c) Zero velocity
   d) Zero acceleration

8. If the magnitude of the sum of two vectors is equal to the magnitude of the difference between them, the angle between the vectors is:
   a) 0°
   b) 45°
   c) 90°
   d) 180°

9. A particle is moving in a circle of radius 'r' with constant speed 'v'. The magnitude of the change in its velocity after it has described an angle of 60° is:
   a) v
   b) v√2
   c) v√3
   d) 2v

10. The component of vector A = 2î + 3ĵ along the vector B = î + ĵ is:
    a) 5/√2
    b) 5
    c) √2 / 5
    d) 1/√2

Answer Key for MCQs:

1. d) Acceleration
2. a) (î - ĵ) m/s (Δr = r₂ - r₁ = (2î - 2ĵ) m; vavg = Δr / Δt = (2î - 2ĵ) / 2 = î - ĵ)
3. c) 100 m (R = (v₀²/g) sin(2θ₀). R₁₅ = (v₀²/g) sin(30°) = 50. R₄₅ = (v₀²/g) sin(90°). R₄₅ / R₁₅ = sin(90°) / sin(30°) = 1 / (1/2) = 2. So, R₄₅ = 2 * R₁₅ = 2 * 50 = 100 m)
4. c) Speed
5. c) 90° (A ⋅ B = |A||B| cos θ. A ⋅ B = (1)(1) + (1)(-1) = 1 - 1 = 0. Since magnitudes are non-zero, cos θ = 0, so θ = 90°)
6. b) 6 km/h (Boat velocity vb and river velocity vr are perpendicular. Resultant v = vb + vr. |v|² = |vb|² + |vr|². 10² = 8² + |vr|². 100 = 64 + |vr|². |vr|² = 36. |vr| = 6 km/h)
7. b) Minimum speed (Vertical component is zero, only horizontal component v₀x remains)
8. c) 90° (|A + B|² = |A - B|². A² + B² + 2AB cos θ = A² + B² - 2AB cos θ. 4AB cos θ = 0. Since A, B ≠ 0, cos θ = 0, so θ = 90°)
9. a) v (Δv = v₂ - v₁. |Δv| = √(v₁² + v₂² - 2v₁v₂ cos θ). Here v₁=v₂=v, θ=60°. |Δv| = √(v² + v² - 2v² cos 60°) = √(2v² - 2v²(1/2)) = √(2v² - v²) = √v² = v. Alternatively, use vector diagram - forms an equilateral triangle.)
10. a) 5/√2 (Component of A along B is A ⋅ B̂ = A ⋅ (B / |B|). |B| = √(1² + 1²) = √2. A ⋅ B = (2)(1) + (3)(1) = 5. Component = 5 / √2)

Make sure you understand the concepts behind these formulas and practice solving numerical problems based on them. Pay special attention to projectile motion and uniform circular motion, as they are frequent topics in exams. Good luck!