Motion in Two Dimensions

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Definition of Motion in Two Dimensions

The motion along two co-ordinate axis that’s x axis and y axis is called as Motion in Two Dimensions.

Projectile Motion

When an object having any initial velocity and moving under the force of gravity that makes any angle along the horizontal component is called as projectile motion.

Assumptions for projectile motion

  • The acceleration due to gravity is constant over the range of motion and is directed downward.
  • The effect of air resistance is negligible
  • The rotation of earth does not affect the motion.

Motion along x-axis

Motion along y-axis

Acceleration a = 0Ay=-g
Velocity v = 0
Vy= vay –
gt
Displacement x = vax tY=vay -1/2gt2

Maximum Height of the Projectile

The maximum height of a projectile when the vertical component of the velocity become 0

vy =v0sinθ –gt=0

Suppose t=T will be the time when the vertical component of a velocity reduced to 0. Putting

vy =0 and t=0 in the above equation, we get

T= v0y/g

When T is the half of the total time

h= v0yT – 1/2 gT2

Substituting for T

h= v0y/(v0y/g) – 1/2 g(v0y/g)2

h= (v0y)2/g– 1/2 (v0y)2/g

h= 1/2 (v0y)2/g

substituting for v0y

h=1/2g v02Sin2 θ

Range of the Projectile

The horizontal distance from origin (x=0,y=0) to the point where the projectile returns (X=R,Y=0) is called the range of the projectile and is represented by R. X=R; where t = 2T

x= v0xt

R = 2/g v0x x v0y

R = 2/g (v0x)2  sinθ cosθ

where 2sinθ cosθ = sin2θ

R= (v0x)2 sin2θ/g

Maximum range is where angle is 45 degrees

Rmax= (v0x)2 /g where θ =45

Projectile Trajectory

The path followed by projectile is called as trajectory.

Y = v0yt – 1/2 gt2

Y = v0 Sin θt – 1/2 gt2

x= v0xt

t= x/v0x=x/ v0x cosθ

Substituting for t in

Y = v0yt – 1/2 gt2

we get

Y = v0 Sinθ(x/ v0x cosθ)  – 1/2 g(x/ v0x cosθ)2

Y= xtanθ -1/2g(x2/ v0x2 cos2θ)

For a given value of projection angle θ and initial velocity of a projectile  g, sinθ and cosθ is constant

a=tanθ

b= g/v0x2 cos2θ   putting a and b in Y gives

Y = ax-1/2bx2

Uniform Circular Motion

When an object moves in a circular path is a constant velocity then this type of motion is called as circular motion.

Angular Displacement

The angle body produce during the circular motion is called as angular displacement. It is measures as radian.

θ=S/r

S=rθ

1 radian = 57.3°

Angular Velocity

The change in motion in displacement in  a unit time or 1 second is called as angular velocity. The direction of angular velocity ϖ vector lies in the axis of rotation.

ϖ =θ/t

Angular Acceleration

When the angular velocity changes with respect to time, an angular acceleration produced.

OR

The rate of change of angular velocity  with respect to time defines angular  acceleration.

αav= (ϖ2– ϖ1)/(t2-t1) = Δ ϖ/ Δt

Relation between angular and linear quantities

Relation between linear and angular displacement is The relation between angular and linear velocity is S=rθ

The relation between linear and angular acceleration is v=rϖ

Tangential Velocity α=ra

In a circular motion when the body has linear acceleration as well as arc velocity that is tangent to the circular path is called as tangential velocity.

Time Period

The time required for one complete revolution of motion is called as time period. The period is denoted as T. The greater the angular velocity will be the shorter time required for one complete revolution. Therefore,

T= 2π/ ϖ =2π/2πf=1/f

Centripetal Acceleration

When the object is moving in a circular path and directed toward the circle. It produces acceleration due to the continuous change in the direction of velocity is called  as centripetal acceleration.

The centripetal is a Greek word.

Centripetal Force

The force that acts toward center during the circular motion of a body is called as centripetal force.

Explanation

Consider a ball of mass m tied to a string of length r is whirled with a constant speed in circular orbit with the velocity v. According to the first law of motion  The inertia of ball tends to maintain motion in straight line path. The force acting on a circular motion is c According to the 2nd law of motion Fc.

Fc=mac

Substituting

ac=v2/r

Fc=mv2/r

Fc=mr2 ϖ2 /r

Fc=mrϖ2

This is the relation of centripetal force

Examples

  • A stone attached to the end of a string is whirled in a circle, the tension T in the string provide the centripetal force that constrains the stone to a circular path.
  • In a racing car moving around a curved track the friction at the wheels provides the centripetal force. If the friction “breaks” or not sufficient on the turning car, the car skids off the track.
  • An electron revolves in circular path around a nucleus has electric force which provides the centripetal force.

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