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CIRCULAR MOTION õ As 2 mv F 0 r → = ≠ , so the body is not in equilibrium and linear momentum of the particle moving on the circle is not conserved. However, as the force is contral, i.e., 0 = τ → , so angular momentum is conserved, i.e., ≠ → p constant but → L = constant õ The work done by centripetal force is always zero as it is perpendicular to velocity and hence displacement. By work-energy theorem as work done = change in kinetic energy ∆K = 0 So K (kinetic energy) remains constant e.g. Planets revolving around the sun, motion of an electron around the nucleus in an atom SPECIAL POINTS õ In one dimensional motion, acceleration is always parallel to velocity and changes only the magnitude of the velocity vector. õ In uniform circular motion, acceleration is always perpendicular to velocity and changes only the direction of the velocity vector. õ IN the more general case, like projectile motion, acceleration is neither parallel nor perpendicular to figure summarizes these three cases. õ If a particle moving with uniform speed v on a circle of radius r suffers angular displacement θ in time ∆t then change in its velocity. 2 1 v v v → → → ∆ = ∆ − ∆ 1 1 ˆ v v i → → = 2 2 2 ˆ ˆ v vc osi v sin j → → → = θ + θ 2 2 2 1 ˆ ˆ v ( v cos v ) i v sin j → → → → ∆ = θ − + 2 2 2 2 1 | v | ( vcos v ) v sin → → → → ∆ = θ − + 2 2 | v | 2v 2vc os → ∆ = − θ = 2 2v ( 1 cos ) − θ = 2 2 2v 2sin 2 θ (Q v 1 = v 2 = v) | v| 2v sin 2 → θ ∆ = EXAMPLE BASED ON UNIFORM CIRCULAR MOTION EXAMPLE BASED ON UNIFORM CIRCULAR MOTION EXAMPLE BASED ON UNIFORM CIRCULAR MOTION EXAMPLE BASED ON UNIFORM CIRCULAR MOTION EXAMPLE BASED ON UNIFORM CIRCULAR MOTION Ex.2 A particle is moving in a circle of radius r centrad at O with constant speed v. What is the change in velocity in moving from A to B ? Given ∠AOB = 40º . Sol. | v | → ∆ = 2v sin40º/2 = 2 v sin 20º X v 2 Y v 1 θ θ θ θ a v(t) (a) a v(t + t) ∆ v(t + t) ∆ v(t) a a a a (c) (b) V(t + t) ∆

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Transcript of 4

CIRCULAR MOTION

As 2mvF 0

r

= , so the body is not in equilibrium and linear momentum of the particle moving on the

circle is not conserved. However, as the force is contral, i.e.,

0=

, so angular momentum is conserved, i.e.,

p constant but

L = constant

The work done by centripetal force is always zero as it is perpendicular to velocity and hence displacement.By work-energy theorem as work done = change in kinetic energy K = 0

So K (kinetic energy) remains constante.g. Planets revolving around the sun, motion of an electron around the nucleus in an atom

SPECIAL POINTS

In one dimensional motion, acceleration isalways parallel to velocity and changes onlythe magnitude of the velocity vector.

In uniform circular motion, acceleration isalways perpendicular to velocity and changesonly the direction of the velocity vector.

IN the more general case, like projectile motion,acceleration is neither parallel nor perpendicular tofigure summarizes these three cases.

If a particle moving with uniform speed v on a circle of radius r suffers angular displacement in timet then change in its velocity.

2 1v v v

= 1 1 v v i

= 2 2 2 v v cos i v sin j = +

222 1

v (v cos v ) i v sin j = +

2 222 1| v | (v cos v ) v sin

= +

2 2| v | 2v 2v cos = = 22v (1 cos ) = 2 22v 2sin 2

(Q v1 = v2 = v)

| v | 2v sin2

=

EXAMPLE BASED ON UNIFORM CIRCULAR MOTIONEXAMPLE BASED ON UNIFORM CIRCULAR MOTIONEXAMPLE BASED ON UNIFORM CIRCULAR MOTIONEXAMPLE BASED ON UNIFORM CIRCULAR MOTIONEXAMPLE BASED ON UNIFORM CIRCULAR MOTION

Ex.2 A particle is moving in a circle of radius r centrad at O with constant speed v. What is the change invelocity in moving from A to B ? Given AOB = 40 .

Sol. | v | = 2v sin40/2 = 2 v sin 20

X

v2

Y

v1

a v(t)

(a)

a v(t + t)

v(t + t)

v(t) a

a a

a

(c)(b)

V(t + t)