How Does Fluid Viscosity Affect Spring Oscillation Period?

[SirR3D]

So there is this ball held on a spring. It's radius R = 0.015m , and density ρ=7800kg/m^3.
It's oscillation period in air is 1.256 seconds and in the fluid it changes to 1.57 seconds.
Find the viscosity of the fluid considering that the drag force obeys Stokes law.

I first found the spring constant k to be k = 2.76.
But how do I find the drag force without knowing the terminal velocity? Or is there a way of solving this otherwise.

Darn I posted this in the wrong section. May someone please move it to the coursework section?
Sorry for the inconvenience

 

[drvrm]

well you have found k of the spring meaning thereby that you have used a relation between time period and k; this relation must have come from some dynamical equation involving forces ...proceed in the same manner using viscous drag then you can go ahead...

 

[SirR3D]

As I said, in the manner I'm trying to solve it I have too many unknowns. I get 6*pi*η*R*v_terminal = mg - ρVg - kx
But I do not know η and v_terminal
Is there any way to solve this without needing to find v_terminal

 

[drvrm]

As I said, in the manner I'm trying to solve it I have too many unknowns. I get 6*pi*η*R*vterminal = mg - ρVg - kx

the above expression i think is not correct as your body is oscillating-not moving with a terminal velocity only downward- the above does not represent the physical situation- try drawing the force diagram and write equation of motionactually one can not go your way as oscillator velocity is not a terminal one as in stokes law- but the forces operating on the body must be written out and viscous drag which opposes the motion may be proportional to speed of the body so its a case of damped oscillator witha damping force due to viscous drag- consult a textbook on it -damped oscillations and this may help

 

[Chestermiller]

You're supposed to assume that the terminal velocity drag force formula applies to the ball at all velocities. This is the approximation that they expect you to make.

Chet

 

[drvrm]

You're supposed to assume that the terminal velocity drag force formula applies to the ball at all velocities. This is the approximation that they expect you to make.

Well i think he should express the drag force as dependent upon the velocity of the body -that gives him a damping factor and that is increasing the time period the coefficient of velocity dependence can give him coefficient of viscosity if he can use the stokes law in expressing the viscous drag force but the new time period must be calculated

I'm trying to solve it I have too many unknowns. I get 6*pi*η*R*vterminal = mg - ρVg - kx
But I do not know η and vterminal

i agree that your formula is something in error; i feel the forces operating are -ky- b(dy/dt) where b is the proportionality constant for drag dependence ; then naturally your solution for oscillator equation which you should write will be modified and naturally new time period will appear there fore you move as the physics goes.

 

[Chestermiller]

Well i think he should express the drag force as dependent upon the velocity of the body -that gives him a damping factor and that is increasing the time period the coefficient of velocity dependence can give him coefficient of viscosity if he can use the stokes law in expressing the viscous drag force but the new time period must be calculated

I didn't say how to solve the problem. I only said that the drag force should be calculated using the Stokes equation, even though the ball is not at terminal velocity during any part of the oscillation. But now that you're asking, I will show specifically how to solve it:

If x represents the upward displacement of the ball relative to the buoyant equilibrium position, then the force balance on the oscillating ball is:

$$\frac{4}{3}\pi\rho R^3\frac{d^2x}{dt^2}=-kx-6\pi\mu R\frac{dx}{dt}$$where $\mu$ is the fluid viscosity and $\rho$ is the density of the ball. This second order homogeneous ODE with constant coefficients can easily be solved for the period of the oscillation.