An object of mass m swings in a horizontal circle on a string of length l that tilts downward at angle θ.

find an expression for the angular velocity ω in terms of g, l and angle θ.

The expression would be ω =

Explanation:

Given that ω is the angular velocity

g is the acceleration due to gravity

L is the length

θ is the angle of downward tilt

For an object we compare the horizontal and vertical component of the forces acting on the body;

For vertical component

T sinθ = mg............1

For the horizontal component

T cos θ = .............2

R is our radius and is  = L cos θ

v = ωR

substituting into equation 2 we have

T cos θ = m(ωR /R

T cos θ=m(ωR ..................3

Now comparing the vertical and the horizontal component we have;

equation 1 divided by equation 3 we have

T sin θ /T cos θ = mg / m(ωR

Tan θ = g / (ωR............4

Making ω the subject formula we have;

(ω = g/  R Tan θ

But R = L cos θ and Tan θ = sin θ/ cosθ

putting into equation 4 we have;

(ω = g /[( L cos θ) x( sin θ/ cosθ)]

(ω = g/ L sinθ

ω =

Therefor the expression for the angular velocity ω in terms of g, L and angle θ would be ω =