CIRCULAR MOTION: Points to Remember

Angular variables


θ = angular position of the particle

ω = angular velocity = d θ/dt = lim∆t→0 ∆θ/∆t

α = angular acceleration = d ω/dt = d²θ/dt²



If the angular acceleration is constant, formulas similar in form to linear formulas can be used to find the angular variables:



θ = ω₀t + ½ αt²
ω = ω₀ + αt
ω² = ω₀² + 2 α θ



Linear variables of circular motion

s = Linear distance traveled by the particle in circular motion
∆s = Linear distance traveled by the particle in circular motion in time ∆t

∆s = r∆θ
Where

r = radius of the circle over which the particle is moving

∆θ = angular displacement in time ∆t

∆s/∆t = r∆θ/∆t
v = r ω

where

v = linear speed of the particle
a_t = rate of change of speed of the particle in circular motion

a_t = dv/dt = rdω/dt = r α



Unit Vectors along the Radius and Tangent on a point on the circle on which the particle is moving.


If x-axis is horizontal and y axis is vertical (normal representation)

e_r = unit vector along the radius at a point on the circle
e_r = i cos θ+j sin θ
e_t = unit vector along the tangent at a point on the circle

e_t = - i sin θ+ j cos θ

Position vector of the point P

If the angular position is θ, x coordinate is r cos θ, and y coordinate is r sin θ.

Hence position vector is r = r(i cos θ + j sin θ)

Differentiating position vector with respect to time we get velocity
v = r ω(-i sin θ + j cos θ)

Differentiating v with respect to time
a = dv/dt = - ω²r e_r + dv/dt e_t



Uniform Circular Motion

• If the particle moves in the circle with a uniform speed (v = constant), it is uniform circular motion.
• In this case dv/dt = 0

 Hence
a = dv/dt = - ω²r e_r

• Acceleration of the particle is in the direction of - e_r , i.e, towards the centre of the circle. The magnitude of the acceleration is

a = ω²r = v²/r (as v = rω)



Nonuniform circular motion

• In nonuniform circular motion, speed is not constant and the acceleration of the particle has both radial and tangential components.

a = - ω²r e_r + dv/dt e_t

• the radial component is a_r = - ω²r = -v²/r
• the tangential component is a_t = dv/dt
• The magnitude of the acceleration

a = √ (a_r² + a_t²)  =  √[( ω²r) ² + (dv/dt) ²]

• The direction of this acceleration makes an angle α with the radius connecting the centre of the circle with the point at that instant.

tan α = (dv/dt)/ (ω²r)


Forces in Circular Motion

• If a particle of mass m is moving along a circle with uniform speed, the force acting on it has to be

F = ma = mv²/r = m ω²r

This force is called centripetal force.


Centrifugal Force

It is equal to the centripetal force

F = mv²/r = m ω²r



Circular Turnings on the Road



When a turn is made, the speed has to be less so that friction can make the vehicle turn in stead of skid

For a safe turn we need to have

 M v²/r ≤ f_s

As f_s≤ μ_sMg

M v²/r ≤ μ_sMg

That means v²/gr ≤ μ_s

v ≤√( μ_sgr) (safety speed)

BANKING OF ROADS



• Instead of relying on friction, road are given an angle to provide centripetal force.

tan θ = v²/gr (angle of banking)

• Safety speed for a banked road with friction

v = [rg(sinθ + μ_s cosθ) / (cosθ - μ_s sinθ)]^1/2.







Conical Pendulum

CONICAL PENDULUM

  

Vertical Circular Motion

• Tension on a string in vertical circular motion

 T = mv²/r + mgcosθ

Maximum Tension

• At bottom most point

 T_max = mv₁²/r + mg

Minimum Tension

• At top most point

 T_min = mv₂²/r - mg

          T_max - T_min = 6mg

Velocity of a particle in vertical circular motion

• At top most point

               v₂ > (rg)^1/2

• At bottom most point

              v₁ > (5rg))^1/2