Accelerating Beaker containing liquid

[andyrk]

When a Beaker is accelerating in the x-direction, the water level gets sloped. Why does this happen? Is this situation analogous to the sloping pendulum in an accelerating cart? Similarly why doesn't the water level change in shape when the beaker is accelerated in y-direction? That is, the surface remains horizontal. Whereas in the case of an accelerating beaker the surface slopes. I think it has something to do with inertia but I think it also involves laws of motion. Can anyone help me understand it? Like in the case of a pendulum hanging from the roof of an accelerating cart, the pendulum aligns at angle θ to the vertical. This occurs because of inertia firstly and secondly so as to fulfil the Newton's law's of motion, i.e the motion is in that direction in which there is force. So as the pendulum is moving in forward direction ( because it is the part of the cart) so the net force on it should also be forward along the direction of acceleration. So that is the reason why the pendulum aligns at an angle θ to the vertical. Because when we resolve the tension along the string it resolves into 2 components, Tsinθ and Tcosθ. The Tcosθ component equals the weight of the body and therefore all forces cancel along the vertical direction. The Tsinθ force is along the forward horizontal direction and there is no force to balance it. So Newton's second law is obeyed, i.e the net force is in the direction of net acceleration (in this case forward). So how to explain the slope of the liquid when the beaker is accelerated in the x-direction? Is the explanation analogous to that of the pendulum in an accelerating cart? If yes then how? I am not able to understand that.

 

[mfb]

Is this situation analogous to the sloping pendulum in an accelerating cart?

Yes.

Where is the y-direction in your coordinate system?

I think it has something to do with inertia but I think it also involves laws of motion.

Right.

So how to explain the slope of the liquid when the beaker is accelerated in the x-direction? Is the explanation analogous to that of the pendulum in an accelerating cart? If yes then how?

You can do the same analysis for the surface: There is no net force perpendicular to the surface (otherwise water would move in that direction and change the surface shape), the vertical forces have to cancel, and the horizontal force has to lead to the acceleration.

 

[andyrk]

Can you explain it a bit quantitatively rather than qualitatively?

Is anyone there? Can anybody explain that using Newton's laws of motion? Just like the bob hanging from the cart?

 

[mfb]

The quantitative part is exactly the same as for the pendulum.
Please don't make multiple posts in a row.

 

[Nickie]

The sloping of the liquid in the beaker when it is accelerated in the x-direction is due to the same principle of inertia that is observed in the pendulum in an accelerating cart. When the beaker is accelerated, the liquid inside it also experiences the same acceleration. However, due to its inertia, the liquid tends to stay at rest and maintain its original position. This results in the liquid sloping towards the direction of acceleration, just like the pendulum aligning at an angle θ to the vertical.

This phenomenon can also be explained using Newton's laws of motion. The liquid in the beaker experiences a net force in the direction of acceleration, just like the pendulum experiences a net force in the direction of the cart's acceleration. This net force causes the liquid to move and form a sloped surface.

In contrast, when the beaker is accelerated in the y-direction, the liquid experiences an equal and opposite force in the opposite direction, thus balancing out and keeping the surface of the liquid horizontal. This is similar to how the pendulum experiences a tension force in the opposite direction to the cart's acceleration, keeping it aligned at an angle θ to the vertical.

Overall, the sloping of the liquid in an accelerating beaker can be explained using the principles of inertia and Newton's laws of motion, just like the sloping of a pendulum in an accelerating cart.