How to Determine the Angle of a Fluid in a Moving Vehicle Using Fluid Mechanics?

[AxiomOfChoice]

Suppose you're driving along a straight, smooth, level stretch of highway with a constant velocity $v$. There is a cup of some fluid in your cup holder. How would you go about finding the angle $\theta$ that the surface of the fluid makes with the bottom of the cup as a function of $v$? And what if you started out with a velocity $v_0$ at time $t=0$ and then accelerated at a constant rate $a$? Can we find $\theta(t)$ without much sweat?

 

[jpreed]

The angle of the liquid in the cup is determined by the acceleration of the car.

If you are traveling at constant velocity the angle of the liquid inside the cup will be zero. Think about when you fly on a plane at 350 mph, the liquid is still flat (unless the plane is slightly tilted).

If you start at zero velocity and begin to accelerate constantly, then the angle of the liquid in the cup will be

$$\theta = \tan^{-1} \left (\frac{a}{g}\right)$$

where $$a$$ is your acceleration and $$g$$ is gravity.

 

[astrokreng]

To find the angle \theta that the surface of the fluid makes with the bottom of the cup as a function of v, we can use the principles of fluid mechanics. First, we need to consider the forces acting on the fluid in the cup. As the car is moving at a constant velocity v, the fluid is also moving at the same velocity. This means that the fluid experiences an acceleration due to the movement of the car. This acceleration creates a force on the fluid, causing it to tilt at an angle \theta.

To determine the value of \theta, we can use the equation for the force on a fluid in a tilted container, which is given by F = mg tan\theta, where m is the mass of the fluid, g is the acceleration due to gravity, and \theta is the angle of tilt. Since we know the values of m and g, we can solve for \theta by rearranging the equation to \theta = arctan(F/mg).

In the case of starting with a velocity v_0 at time t=0 and then accelerating at a constant rate a, the angle \theta will change over time. This is because the fluid will experience a changing acceleration as the car accelerates. To find \theta(t), we can use the same equation as before but substitute the changing acceleration for g. This means that \theta(t) = arctan(F/ma).

Overall, finding the angle \theta that the surface of the fluid makes with the bottom of the cup as a function of v or \theta(t) when accelerating at a constant rate can be done using the principles of fluid mechanics and the equation for the force on a tilted fluid. It may require some mathematical calculations, but with the right equations and values, we can determine the angle without much difficulty.