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https://docs.google.com/document/d/1UEmVvmknDkL43jm8DqWKiZCjQJ5NW5VM/edit?usp=sharing&ouid=109474854956598892099&rtpof=true&sd=true Description of Motion in Two and Three Dimensions Scalars and Vectors, Vector addition and Subtraction, Zero Vector, Scalar and Vector products, Unit Vector, Resolution of a Vector. Relative Velocity, Motion in a plane, Projectile Motion, Uniform Circular Motion. SCALARS AND VECTORS Scalars Scalars are physical quantities which are completely described by their magnitude only. For example: mass, length, time, temperature energy etc. Vectors Vectors are those physical quantities having both magnitude as well as direction and they obeys vector algebra (eg. parallelogram law or triangle law of vector addition). For example: displacement, velocity, acceleration, force, momentum, impulse, electric field intensity etc. TYPES OF VECTOR Axial and Polar Vectors Vectors, which have some starting point or point of application are called polar vectors. E.g., displacement, force but those vectors representing rotational effects and are always along the axis of rotation in accordance with right hand screw rule are axial vectors. E.g., angular velocity, angular acceleration, angular displacement, torque etc. Unit Vector : A vector having unit magnitude is called a unit vector. Thus, unit ˆ V V C H A P T E R THIS CHAPTER COVERS : Scalars and Vectors Types of Vector Vector Addition Vector Subtraction Resolution of Vectors Scalar and Vector product Relative Motion Projectile Motion Circular Motion vector of a vector V is V = = | V | V where | V | = V = Magnitude of V . iˆ, ˆj and kˆ are unit vectors along x, y and z axis respectively. | iˆ |=| ˆj |=| kˆ |= 1 . Unit vector along a direction is unique and have no unit. Coplanar vectors are vectors lying in same plane. I I and J are in the plane of paper so they are Coplaner Collinear vectors are vectors having common line of action. These are of two types Parallel or like (angle between them is 0°) A B A and B are parallel vectors (θ = 0°) Parallel vectors having equal magnitude are known as equal vectors Antiparallel or unlike (angle between them is 180°). C C and D are antiparallel vec Antiparallel vectors of equal megnitude are known as negative vectors of each other. Null vector ( 0 ) : Two opposite vectors added to form a null vector it has zero magnitude. If A and B are two negative vectors then A + B = 0 VECTOR ADDITION vector Triangle Law of Vector Addition : If a and b are two vectors represented as sides of a triangle in same order then other side c in opposite order is the resultant. b a Polygon Law : If a number of vectors are represented as sides of a polygon in same order then the side which closes the polygon in opposite order in the resultant. Vector addition obeys commutative law (A + B = B + A) and associative law (A + B) + C = A +(B + C) Parallelogram Law of Vector Addition : If two vectors having common origin are represented both in magnitude and direction as the two adjacent sides of a parallelogram, then the diagonal which originates from the common origin represents the resultant of these two vectors. The result are listed below: (a) (b) (c) R = A + B . B | R |= (A2 + B 2 + 2AB cos θ)1/ 2 tanα = B sinθ , tanβ = A sinθ A + B cos θ B + A cos θ If | A |=| B |= x (say) , then R = x = 2x cos θ 2 and A θ α = β = 2 i.e., resultant bisect angle between A and B . If | A |>| B | then α < β Rmax = A + B, when θ = 0 and Rmin = |A – B| when θ = 180°. R 2 = A2 + B2, if θ = 90° i.e., A and B are perpendicular. (h) If | A |=| B |=| R | then θ = 120°. If R is perpendicular to A , then cos θ = − A B and A2 + R2 = B2. ⎛ 360 ⎞ For n equal vectors acting at a point such that angle between them are equal ⎜ ⎝ ⎟ , the resultant n ⎠ is zero. VECTOR SUBTRACTION Subtraction of vector B from vector A is simply addition of vector − B Using parallelogram law with A i.e., A − B = A + (−B) Result : R = | A − B |= A2 + B2 − 2A cos θ,tanα = B sin(π − θ) = B sinθ A + B cos(π − θ) A − B cos θ RESOLUTION OF VECTORS Any vector V can be represented as a sum of two vectors P and Q which are in same plane as V = λ P + μQ , where λ and μ are two real numbers. We say that V has been resolved in two component vector λP and μQ along P and Q respectively. Rectangular components in two dimensions : V = V x V y , V = V x iˆ + V y ˆj, V = Y V V x and V y are rectangular component of vector in 2-dimension. V = V cos θ Vy Vy = V sin θ = V cos(90 – θ) Vz = zero. V =V cos θ iˆ + V sin θ ˆj θ O X Vx Rectangular Component in Three Dimension : A vector V is in a space which is making α, β and γ with x-axis, y-axis and z-axis respectively. V = V x + V y + V z 2 2 2 1 | V |= ( V x + V y + V z ) 2 V = V x iˆ + V y ˆj + V zkˆ Vx = V cos α ⇒ cos α = l = x X V V Vy = V cos β ⇒ cos β = m = V Z V = V cos γ ⇒ cos γ = n = Vz z V V = V cos α iˆ + V cosβˆj + v cos γ kˆ l, m, n are called direction cosines of vector V . l 2 + m2 + n2 = cos2 α + cos2 β + cos2 γ = 1, sin2 α + sin2 β + sin2 γ = 2. Unit vector along V = l iˆ + mˆj + nkˆ = cos αiˆ + cosβˆj + cos γkˆ . SCALAR AND VECTOR PRODUCTS Scalar (dot) Product of Two Vectors : The scalar product of two vectors a and b is defined as ρ a . b = ab cos θ ρ cos θ = a . b ab If a and b are perpendicular, then ρ a . b = 0 ρ If θ < 90° , then a .b > 0 and if ρ < 0 , then θ > 90°. Projection of vector a on b is a2 = a .a (a .b) b . b2 iˆ.iˆ = jˆ. = kˆ.kˆ = 1. Scalar product is commutative i.e., a. b = b.a . Vector Product of two Vectors : → → Mathematically, if θ is the angle between vectors A and B , then A ×B → → A×B = AB sinθ nˆ → → The direction of vector × …(i) is the same as that A θ of unit vector nˆ . It is decided by any of the following two rules : B (a) Right handed screw rule : Rotate a right handed screw from vector → → to B through the smaller angle between them; then the direction of motion of screw gives the direction of vector → → (Fig. a) A×B Right hand thumb rule : Bend the finger of the right hand in such a way that they point in the direction of rotation from vector → to → through the smaller angle between them; then the thumb points in the → → direction of vector A×B (Fig. b) Some Important Points : The cross product of the two vectors does not obey commutative law. As discussed above → → → → → → → → A × B = − (B × A) i.e., A× B ≠ (B × A) The cross product follows the distributive law i.e., → → → → → → → A ×(B + C ) = A× B + A × C The cross product of a vector with itself is a NULL vector i.e., → → A × A = (A) (A) sin0° nˆ = 0 The cross product of two vectors represents the area of the parallelogram formed by them, (Figure., shows a parallelogram PQRS whose adjacent sides PQ and PS are represented by vectors → → A and B respectively. → → Now, area of parallelogram = QP × SM = QP⋅AB sin θ. Because, the magnitude of vectors A×B is AB sin θ, hence cross product of two vectors represents the area of parallelogram formed by it. It is worth → → → → noting that area vector A×B acts along the perpendicular to the plane of two vectors A and B . In case of unit vectors iˆ, ˆj, kˆ, we obtain following two important properties: (a) iˆ × iˆ = jˆ × ˆj = kˆ × kˆ = (1) (1) sin 0° (nˆ) = 0 (b) iˆ × ˆj = (1) (1) sin 90° (kˆ) = kˆ where, kˆ is a unit vector perpendicular to the plane of iˆ and ˆj in a direction in which a right hand screw will advance, when rotated from iˆ to ˆj Also, − ˆj × iˆ = −(1) (1) sin 90° (−kˆ) = kˆ Similarly, ˆj × kˆ = −kˆ × ˆj = iˆ and kˆ × iˆ = − iˆ × kˆ = ˆj Cross product of two vectors in terms of their rectangular components : → → ˆ ˆ ˆ ˆ ˆ ˆ A× B = (Axi + Ay j + Azk ) ×(Bxi + By j + Bzk ) = (Ay Bz − Az By ) iˆ + (Az Bx − Ax Bz ) jˆ + (Ax By − Ay Bx ) kˆ iˆ ˆj kˆ = Ax Ay Az Bx By Bz RELATIVE MOTION IN TWO DIMENSIONS Relative velocity : Velocity of object A w.r.t. object B is v AB = v A − vB , vBA = v B − v A Direction of Umbrella : A person moving one straight road has to hold his umbrella opposite ot direction of relative velocity of rain. The angle θ is given by tanθ = v M v R with vertical in forward direction. Umbrella Closest approach : Two objects A and B having velocities vA figure and vB at separation x are shown in The relative velocity of A with respect to B is given by -vB v AB = v A v B A tanβ = v A v B The above situation is similar to figure given below y is the distance of closest approach. Now, sinβ = y x ⇒ y = x sinβ y = = Crossing a river : v = velocity of the man in still water. θ = angle of which man swims w.r.t. normal to bank such that v = – v sin θ, v = v cos θ Time taken to cross the river is given by t = d = d Velocity along the river vy v cos θ v sin θ A u x ′ = u − v sinθ Distance drifted along the river D = t v ′ D d v cos θ (u − v sinθ) Case I : The Minimum time to cross the river is given by τmin = v (when cos θ = 1, θ = 0°, u ⊥ v) Distance drifted is given by D = d × u v Case II : To cross the riven straight u drift D = 0 ⇒ u – v sin θ = 0 sinθ = u v ⇒ (v > u) Time to cross the river straight across is given by t d = v cosθ PROJECTILE MOTION An object moving in space under the influence of gravity is called projectile. Two important cases of interest are discussed below : Horizontal projection : A body of mass m is projected horizontally with a speed u from a height h at the moment t = 0. The path followed by it is a parabola. It hits the ground at the moment t = T, with a velocity v such that T = t = 0 x-axix ρ ˆ ˆ y v = = ui + gTj The position at any instant t0 is given by x = ut0 y = 1 gt 2 2 0 gx 2 y 2u 2 The velocity at any instant t0 is given by v 0 = uiˆ + gt 0 ˆj Oblique projection : A body of mass m is projected from ground with speed u at an angle θ above horizontal at the moment t = 0. It hits the ground at a horizontal distance R at the moment t = T. Time of flight T = 2uy g u2 = 2u sinθ g u2 sin2 θ Maximum height H = y = 2g 2g 2uxuy u 2 sin 2θ Horizontal range R = ux ×T = g = g Equation of trajectory; gx 2 y = x tan θ − 2u 2 cos2 θ or y = x tan θ⎛1 − x ⎞ ⎝ R ⎠ Instantaneous velocity v = gt v sinβ and direction of motion is such that, u cos θ tanβ = tan θ − u cosθ v cosβ (a) v = cosβ [Θ Horizontal component is same every where] v sinβ = u sinθ – gt When v (velocity at any instant ‘t’) is perpendicular to u (initial velocity) ⇒ β = 90° – θ (i) (ii) v = u cos θ cos(90° − θ) t u g sinθ = u cot θ Applications : The height attained by the particle is largest when θ = 90°. In this situation, time of flight is maximum and range is minimum (zero). The horizontal range is same for complimentary angles like (θ, 90 – θ) or (45 + θ, 45 – θ). It is maximum for θ = 45°. When horizontal range is maximum, H = Rmax 4 When R is range, T is time of flight and H is maximum height, then gT 2 (a) tanθ = 2R (b) tan θ = 4H R If A and B are two points at same level such that the object passes A at t = t1 and B at T = t2, then y (i) (ii) T = 2u sinθ = t g 1 h = 1 gt t T x t2 2 1 2 Average velocity in the interval AB is vav = ucosθ [Θ vertical displacement is zero] CIRCULAR MOTION An object of mass m is moving on a circular track of radius r. At t = 0, it was at A. At any moment of time ‘t’, it has moved to B, such that ∠ AOB = θ . Let its speed at this instant be v and direction is along the tangent. In a small time dt, it moves to B′ such that ∠ B′OB = d θ . The angular displacement vector is dθ = dθkˆ The angular velocity vector is ρ = dθ kˆ . dt At B′, the speed of the object has become v + dv. The tangential acceleration is a = dv t dt v 2 The radial (centripetal acceleration) is ac = r The angular acceleration is α = dω dt Relations among various quantities. = ω2r x 1. v = ω× r ρ dv ρ dr dω ρ 2. a = dt = ω× dt + dt × r = ac + at 3. a ρ ρ c = ω× v 4. a ρ ρ t = α × r