CHAPTER 3-MOTION IN A PLANE

INTRODUCTION-

In one dimension only two direction are possible .But in order to describe motion of a object in 2D or 3D we use vector. In this chapter we will learn language of vector. Motion in a plane is called 2D motion. In 2D geometry have 6 planes XY, YZ, ZX, -XY ,-YZ- ,-ZX.

SCALAR QUANTITY AND VECTOR QUANTITY-

Which physical quantity have only magnitude is called scalar quantity whereas which physical quantity have both magnitude and direction is called vector quantity. In scalar quantity ,we usually simple mathematical operation like addition, subtraction, multiplication and division. But in vector quantity we use triangle law of vector addition , parallelogram law of vector addition etc. Vector is denoted by v (in bold letter) or an arrow on top of letter.

POSITION AND DISPLACEMEN VECTOR –

In a cartesian plane , if a body move A to B ,we have a line AB on it. The length of vector is represented by magnitude of line AB and the direction of vector is equal to the direction of line AB. If a body place in a plane with respect to origin of coordinate is called position vector .If a body move A to B, the vector AB(with tail with at A and tip at B) is called displacement vector. The magnitude of displacement is either less or equal to the path length of an object between two point. Unit vector is used to specify direction and it is represented by putting a cap (^) over a quantity(ex. unit vector in x-direction is î etc.)

EQUALITY OF VECTOR-

Two vectors A and B is said to be equal when only if they have same magnitude and same direction ,no matter where it placed in a plane.

MULTIPLICATION OF VECTOR BY REAL NUMBER

Multiplication of vector V with a positive number π is |V π| = π | V | if π > 0.

If A is multiplied by 2 ,then result is 2A.

ADDITION AND SUNTRACTION OF VECTORS – GRAPHICAL METHOD-

Every vector are arranged head to tail, this graphical method is called head to tail method.

Triangle law of vector addition states that if two vectors are arranged tip to tail, then the third side of triangle(from start to end) represent their resultant(R) (resultant are also called result).

After proof(*proof in question section*) we get a formula ,R =under root(A² + B² + 2ABcosθ) and angle tan alpha (α), tan α = (B sinθ) / (A + B cosθ).or α = tan^-1 [B  sinθ) / (A + B cosθ ].

The parallelogram law of vector addition states that if two vectors are represented as adjacent sides of a parallelogram, their resultant vector is represented by the diagonal that connects the opposite corners. After proof(*proof in question section*) we get A + B = B + C = R .

When we subtract vector A form vector B then there are no additional rule, we use property of vector addition like negative vector and triangle rule .

A vector with zero magnitude and direction is called null vector or zero vector. A vector which is same in direction , may and may not same in magnitude is called parallel vector and which have in opposite direction is also called antiparallel vector. If a vector lie in same plane is called coplanar vector. And a vector in negative direction is called negative vector.

RESOLUTION OF VECTORS –

Magnitude of vector are represented by these formulas (after derivation) we get A= A_Xi^ + A_Yj^ +A_Zk^ .

Magnitude of vector  A is A= under root(A²_X + A²_Y  + A²_Z )

Position of vector R can be expressed by as R = xi^{^}+ yj^{^} + zk^{^}

VECTOR ADDITION – ANALYTICAL METHOD –

After derivation(derivation in question) we get a formula –R=(Ax+ Bx)i^ + (Ay+By)j^ + (Az+Bz)k^

MOTION IN A PLANE-

We describe motion in 2D by using vector –

Position vector and Displacement – If position vector r of an object p is located in x-y plane ,it is given by r = xi^ + yj^ in addition and r= r₂ – r₁.

Velocity- The average velocity of an object is ratio of displacement with respect to time v = dxi^/dt + dyj^/dt or v = v_x i^{^} + v_yj^{^} . The instantaneous velocity is given by limiting value of the average velocity as the time interval as approaches zero : v = lim dr / dt. The direction of velocity at any point on the path of object is tangential to the path at that point and is in the direction of motion. Where V_x = dx/dt ,v_y = dy/dt.

Acceleration – The average acceleration of an object for time interval(dt) moving in x-y plane is change in velocity divided by the time interval :a = a_xi^ + a_yj^{^} ,where a_x=dv_x/dt  ,a_y=dv_y/dt.

In one dimension ,the velocity and the acceleration of an object are always along the same straight line(either in same or opposite direction). However, for motion in 1D or 2D, velocity and acceleration vectors may have and any angle between 0 degree and 180 degree between them .

MOTION IN A PLANE WITH CONSTANT ACCELERATION –

Suppose an object moving with constant acceleration a in x-y plane ,then

v_x=u_x+a_xt, v_y=u_y+a_yt

 s_x=u_xt +1/2a_xt², s_y=u_yt+1/2a_yt²

v²_x=u²_x+2a_x s_x, v²_y=u²_y+2a_ys_y

PROJECTILE MOTION –

It is the curved path an object follows when it is thrown or projected near Earth surface ,it influence only by gravity and its initial velocity. The path of projectile is called trajectory.

The equation of path of projectile is – Y = (tan θ₀) x – g/2(v₀cos θ₀)*x²

Time of maximum height – Half flight time – t=usinθ/g. Full flight time –t= 2usinθ/g

Maximum height of a projectile – H=(u²sin²θ)/2g

Horizontal range of a projectile – R=(u²sin2θ)/g

UNIFORM CIRCULAR MOTION-

When an object follow a circular path with constant speed is uniform circular motion.

when the speed of an object is constant in circular path, then it change it s’ direction at every point .Centripetal acceleration (ac) = V²/R  ,in different case ac=rω²  and ac=4π²f²r in different case. Where f=frequency ,r=radius, v=velocity, ω=angular displacement(omega).

These all are the notes of chapter 3 in physics. And after some time you get important questions and NCERT solutions HERE. *#THANKS FOR VISITING, VISIT AGAIN#* 😊