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Answers
Answer:
hopefully this is helpful
Explanation:
When a body falls in a viscous medium, it carries with its layers of fluid which are in body's contact where as the layers of fluid in contact with the stationary surface remain almost at rest. The layers of fluid destroys the relative motion and motion of the body is thus opposed. The viscous drag increases with velocity of the body till viscous drag and upthrust of the body are together equal to the weight of the body which acts downwards.
When there is no net force, the body moves with the uniform velocity. This velocity is called terminal velocity.
Stoke showed that the retarding force F due to viscous drag for a spherical body of radius r that moves with a velocity v in a fluid, with coefficient of viscosity η, is given as :
F=6πηrv
This expression is known as Stoke's Law.
Derivation with help dimensions :
F α v, velocity
F α r, radius of the body;
F α η, coefficient of viscosity of the fluid.
⇒F=Aη
a
r
b
v
c
where A is a constant with no dimensions. Putting the dimensions,
[MLT
−2
]=[ML
−1
T
−2
]
a
[L]
b
[LT
−1
]
c
or [MLT
−2
]=[M
a
L
−a+b+c
T
−a−c
]
Comparing the two sides, we get
a = 1, b = 1 and c = 1
∴F=Aηrv
A was determined to be 6π.
∴F=6πηrv
Derivation for terminal velocity :
Let p be the density of the body and ρ be the density of the medium. Then,
Weight of body =
3
4
πr
3
pg ........(11.4)
Upthrust of body =
3
4
πr
3
σg ........(11.5)
Resultant force (downwards) =
3
4
πr
3
(p−σ)g
In equilibrium,
6πηrv=
3
4
πr
3
(p−σ)g ......(11.6)
⇒v=
9
2
r
2
η
(p−σ)
g .....(11.7)
This is the formula for terminal velocity.