Motion in a Plane - Class 11 Physics

Motion in a Plane

Overview: This chapter extends motion to two dimensions using Vectors. It covers vector algebra, projectile motion, and uniform circular motion.

1. Scalars and Vectors

Scalars: Quantities with magnitude only (e.g., Mass, Time, Speed).
Vectors: Quantities with both magnitude and direction, and which obey triangle law of addition (e.g., Displacement, Velocity, Force).

Types of Vectors

Unit Vector: A vector of unit magnitude used to specify direction. denoted by â. |â| = 1.
Null Vector: A simple point with zero magnitude and arbitrary direction.
Equal Vectors: Same magnitude and same direction.

2. Vector Operations

Addition of Vectors

Triangle Law: If two vectors are represented by two sides of a triangle in sequence, their sum is the third side in opposite order.

Parallelogram Law: If two vectors are adjacent sides of a parallelogram, their resultant is the diagonal starting from the common point.

R = √(A² + B² + 2AB cosθ)

tan(β) = (B sinθ) / (A + B cosθ)

Resolution of Vector

Splitting a vector into perpendicular components.

A = A_xî + A_yĵ

A_x = A cosθ, A_y = A sinθ

3. Product of Vectors

Scalar (Dot) Product

Returns a scalar quantity.

A ċ B = AB cosθ

Example: Work = F ċ d

Vector (Cross) Product

Returns a vector quantity perpendicular to the plane of A and B.

A × B = (AB sinθ) n̂

Example: Torque = r × F

4. Motion in a Plane

Position Vector: r = xî + yĵ

Velocity: v = v_xî + v_yĵ

Acceleration: a = a_xî + a_yĵ

Motion with Constant Acceleration

Can be treated as two independent 1D motions along x and y axes.

x = u_xt + ½a_xt²

y = u_yt + ½a_yt²

5. Projectile Motion

An object thrown into space under gravity alone. (a_x = 0, a_y = -g).

Equation of Trajectory (Path is Parabola)

y = x tanθ - (gx²) / (2u² cos²θ)

Key Formulas

Time of Flight (T): 2u sinθ / g
Max Height (H): u² sin²θ / 2g
Horizontal Range (R): u² sin(2θ) / g

Max range is at θ = 45°.

6. Uniform Circular Motion

Motion in a circle at constant speed. Velocity direction changes continuously.

Centripetal Acceleration

Acceleration directed towards the center.

a_c = v² / R = ω²R

Angular Variables

Angular Velocity (ω): dθ/dt
Relation: v = ωR
Frequency Formula: v = 2πR / T

Numericals - Motion in a Plane

Numericals

Resultant Vector

Q1. Two forces of 3N and 4N act perpendicular to each other. Find the magnitude and direction of the resultant.

A = 3N, B = 4N, θ = 90°

R = √(A² + B² + 2AB cos90°)

R = √(3² + 4² + 0)

R = √(9 + 16) = √25

R = 5 N

Direction tanβ:

tanβ = (B sin90) / (A + B cos90)

tanβ = 4/3 → β = 53°

53° with the 3N force.

Unit Vector

Q2. Find the unit vector in the direction of vector A = 3i + 4j.

Unit vector â = A / |A|

|A| = √(3² + 4²) = 5

â = (3i + 4j) / 5

â = 0.6i + 0.8j

Scalar Product

Q3. Find the angle between two vectors A = i + j and B = i - j.

Use Dot Product: A ċ B = |A||B| cosθ

A ċ B = (1)(1) + (1)(-1) = 0

|A||B| cosθ = 0

Since magnitudes are non-zero:

cosθ = 0

θ = 90°

Projectile Motion

Q4. A cricket ball is thrown at a speed of 28 m/s in a direction 30° above the horizontal. Find the maximum height.

u = 28 m/s, θ = 30°

H = (u² sin²θ) / 2g

sin 30° = 0.5

H = (28 × 28 × 0.5 × 0.5) / (2 × 9.8)

H = (784 × 0.25) / 19.6

H = 196 / 19.6

H = 10 m

Rain Man Problem

Q5. Rain is falling vertically at 35 m/s. A woman rides a bicycle at 12 m/s East to West. What is the direction of relative velocity?

v_r = 35 (down), v_w = 12 (West)

We need velocity of rain w.r.t woman: v_rw = v_r - v_w

This is vector addition of v_r and -v_w.

tanθ = v_w / v_r

tanθ = 12 / 35

θ = tan^-1(0.343)

Direction is approx 19° with vertical.

River Swimmer

Q6. A swimmer can swim at 4 km/h in still water. River flows at 3 km/h. Find time to cross 1 km river straight across.

To cross straight, he must head upstream.

v_res = √(v_s² - v_r²)

v_res = √(4² - 3²) = √7 km/h

Time = Width / v_res

Time = 1 / √7 hr

Circular Motion

Q7. An aircraft executes a horizontal loop of radius 1 km with a steady speed of 900 km/h. Calculate centripetal acceleration.

R = 1000 m

v = 900 km/h = 900 × (5/18) = 250 m/s

a_c = v² / R

a_c = (250 × 250) / 1000

a_c = 62500 / 1000

a_c = 62.5 m/s²

Range of Projectile

Q8. For what angle of projection are the horizontal range and maximum height equal?

R = H

(u² sin2θ) / g = (u² sin²θ) / 2g

2 sinθ cosθ = (sin²θ) / 2

4 cosθ = sinθ

tanθ = 4

θ = tan^-1(4) ≈ 76°

Vector Cross Product

Q9. Find the torque of force F = 2i + j acting at position r = 3i.

Torque τ = r × F

τ = (3i) × (2i + j)

τ = 6(i×i) + 3(i×j)

i×i=0, i×j=k

τ = 0 + 3k

τ = 3k Nm

Angular Speed

Q10. Calculate the angular speed of the second hand of a clock.

Time period of second hand T = 60 s

ω = 2π / T

ω = 2π / 60

ω = π / 30 rad/s

ω ≈ 0.105 rad/s

Formulas & Facts - Motion in a Plane

Equations & Formulas

Concept	Formula
Resultant Magnitude	R = √(A² + B² + 2AB cosθ)
Resultant Direction	tanβ = (B sinθ) / (A + B cosθ)
Dot Product	A ċ B = AB cosθ
Cross Product	\|A × B\| = AB sinθ
Proj. Max Height	H = (u² sin²θ) / 2g
Proj. Range	R = (u² sin2θ) / g
Proj. Time Flight	T = (2u sinθ) / g
Centripetal Acc.	a_c = v²/R = ω²R
Angular Velocity	ω = 2πν = 2π/T
River Crossing (Shortest)	sinθ = v_r/v_s

50 NEET Facts

Key points for Motion in a Plane.

1. Scalar vs Vector Force, Velocity, Displacement, Torque are Vectors. Speed, Mass, Energy, Pressure are Scalars.

2. Current Scalar? Electric current has magnitude and direction, but it is scalar because it does not obey vector laws of addition (it follows Kirchhoff's laws/algebraic sum).

3. Rotation Effect A vector does not change if it is displaced parallel to itself. It changes if rotated.

4. Minimum vectors for Zero Minimum number of coplanar vectors of equal magnitude whose sum can be zero is 2. If unequal, minimum is 3.

5. Non-coplanar vectors Minimum number of non-coplanar vectors whose sum can be zero is 4.

6. Max Resultant Resultant of two vectors is maximum when angle between them is 0° (A+B).

7. Min Resultant Resultant is minimum when angle is 180° (|A-B|).

8. Perpendicular Vectors Resultant is √(A²+B²) and dot product is zero.

9. Orthogonal Unit Vectors i ċ i = 1, i ċ j = 0. i × i = 0, i × j = k.

10. 45 Degree Projectile At 45°, Range is maximum (u²/g) and H = R/4.

11. Complementary Angles For same initial speed, Range is same for projection angles θ and (90-θ). But heights are different.

12. Vertical Velocity at Top For a projectile, vertical component of velocity at Hmax is zero. Horizontal component remains u cosθ.

13. Kinetic Energy at Top At highest point, KE is not zero. It is K cos²θ (since only horizontal velocity exists).

14. Acceleration in Projectile Acceleration is constant throughout the motion (g acting downwards). It is never zero.

15. Angle of velocity changes Velocity vector is tangent to the path. Angle changes from +θ to -θ.

16. Circular Motion Velocity Linear velocity v is always tangent to the circle. v = ω × r (Vector product).

17. Uniform Circular Motion Acc In UCM, speed is constant but velocity changes direction. Hence, acceleration exists (Centripetal). Tangential acceleration is zero.

18. Work Done in UCM Work done by centripetal force is zero because force is perpendicular to instantaneous displacement (velocity).

19. Non-Uniform Circular Motion Both centripetal and tangential acceleration exist. Net a = √(a_c² + a_t²).

20. Tangential Acceleration a_t is responsible for changing the magnitude of velocity (speed).

21. Centripetal Acceleration a_c is responsible for changing the direction of velocity.

22. Angular Acceleration α = dω/dt. Direction is along the axis of rotation (Right hand rule).

23. Pseudo Force In a rotating frame (non-inertial), centrifugal force acts outwards. Magnitude mv²/r.

24. River Boat Shortest Path To cross via shortest path, swimmer must swim at angle > 90° to downstream. Valid only if v_swimmer > v_river.

25. Drift If swimmer heads straight across (shortest time), he will drift downstream by distance x = v_river × Time.

26. Rain Umbrella To protect from rain, hold umbrella in direction of relative velocity of rain with respect to man.

27. Area of Parallelogram Magnitude of A × B gives the area of the parallelogram formed by vectors A and B.

28. Condition for Collinearity Two vectors A and B are collinear if A = kB or if their cross product is zero.

29. Division by Vector Division of a vector by a vector is not defined.

30. Position Time Graph 2D Trajectory is the plot of y vs x. It is not x vs t.

31. Horizontal Projectile Dropped from height h with horizontal vel u. Time to fall depends only on h (t = √(2h/g)). Range = u × t.

32. Velocity Impact For ground-to-ground projectile, speed of impact = speed of projection (if air resistance neglected).

33. Radius of Curvature For projectile at top, R = u² cos²θ / g. At start, R = u² / (g cosθ).

34. Change in Momentum In projectile motion (full flight), change in momentum is 2mu sinθ (vertical). Horizontal momentum constant.

35. Angular Momentum Projectile About point of projection, Angular Momentum is not constant (Torque by gravity acts).

36. Banking of Roads To prevent skidding at turns, roads are banked. tanθ = v²/rg.

37. Bending of Cyclist Cyclist bends inwards by angle θ = tan^-1(v²/rg) to provide centripetal force.

38. Commutative Laws Vector addition is commutative (A+B = B+A). Dot product is commutative (A.B = B.A). Cross product is anti-commutative (AxB = -BxA).

39. Distributive Law Dot and Cross products distribute over addition. A.(B+C) = A.B + A.C.

40. Lami's Theorem For 3 concurrent forces in equilibrium: F1/sinα = F2/sinβ = F3/sinγ.

41. Zero Resultant If N vectors of equal magnitude act at a point with angle 2π/N between adjacent ones, resultant is zero.

42. Dot Product Sign Positive if 0 < θ < 90. Negative if 90 < θ < 180. Zero if θ=90.

43. Projectile on Incline Range on inclined plane is max when projection angle bisects angle between vertical and incline.

44. Relative Motion 2D If A and B move with same velocity vector, relative velocity is zero. They stay at constant distance.

45. Minimum speed projectile Minimum speed is at highest point (u cosθ).

46. Average Velocity Projectile Over total flight: Horizontal u cosθ. Vertical 0. Average is u cosθ.

47. Time of Flight independence T depends on vertical component u sinθ. Two projectiles with same u_y have same T, regardless of u_x.

48. Max height independence H depends on vertical component. Same u_y means same H.

49. Rate of change of Speed In UCM it is zero. In projectile, it is g sinθ (component along path).

50. Conical Pendulum Time period T = 2π √(L cosθ / g). Similar to circular motion physics.

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