Question

a water droplet falling through the air can oscillate with some angular frequency that depends on its surface tension, density, and radius. the surface tension may be interpreted as the energy per unit area of surface of the drop. if a certain drop oscillates with angular frequency $\omega,$ what is the oscillation angular frequency of a drop with half of the first drop's radius?

Answer 1

The oscillation angular frequency of a drop with half of the first drop's radius is 4ω

What is surface tension?

Surface tension is the tension force exerted on an object by the surface of a liquid.

What is angular frequency?

Angular frequency is the frequency of oscillation of a rotating object. It is given in rad/s.

What is the oscillation angular frequency of a drop with half of the first drop's radius?

Given that

• the angular frequency of the drop is ω and
• radius r.

Since the energy of the drop is conserved, using the law of conservation of angular momentum, we have

Iω = I'ω' where

• I = initial rotational inertia of droplet = mr²
• where m = mass of drop and
• r = initial radius of droplet,
• ω = initial angular frequency of droplet,
• I' = initial rotational inertia of droplet = mr² where
• m = mass of drop and
• r' = final radius of droplet, and
• ω = final angular frequency of droplet

So, Iω = I'ω'

Making ω' subject of the formula, we have

ω' = Iω/I'

ω' = mr²ω/mr'²

ω' = r²ω/r'²

Given that the drop is half of the first drop's radius, r' = r/2

So, ω' = r²ω/r'²

ω' = r²ω/(r/2)²

ω' = r²ω/r²/4

ω' = 4ω

So, the oscillation angular frequency of a drop with half of the first drop's radius is 4ω

Learn more about angular frequency here:

brainly.com/question/28036464

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