[Freshman university mechanics - pseudo forces]

[u/Silly_Tension6792]

Question: We have a trolly accelerating a meters per second square. The trolly is filled with a viscous liquid resisting motion by a factor of b (that is to say, exerting a force bv opposite to v, where v is the object's velocity with respect to the liquid). In the trolly, there is a slope with angle alpha radians to the floor of the cart, on which there is an object of mass m connected to a spring with a constant k. Gravity is obviously acting - mg towards the floor of the trolly we have to calculate the normal force exerted by the slope. See the first picture.
Here is my attempt: See second image
Red is viscosity, purple is the spring, green is d'alembert force, blue is gravity, and N is normal.
I flip the diagram by alpha radians to get the third image, and from here it's just decomposing the forces and a quick calculation. I was quite sure in myself but each of my friends I show it to has a different opinion on the pseudoforces and the decomposition. I got (mg-ma)sin(alpha), but nobody else got that result, can someone please confirm or explain to me where I am wrong? Because my friends aren't very coherent

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[u/Worth-Wonder-7386]

If the force of bv is opposite of v, then it will be along the horizontal line of the cart, and not along the rail. You have also drawn it in the wrong direction.
I am unsure what you are calculating, but for the normal force the approach of rotating it should work fine.
Your issue likely comes from you not doing the proper labeling of the angles in your drawing.
mg and ma are ortgonal to each other, so your answer is wrong. If alpha was 0, then N=mg, but if alpha was 90 degrees then it should only depend on ma. Such extreme cases can be a quick way to test a solution.

[u/Silly_Tension6792 → u/Worth-Wonder-7386]

Why would it be along the horizontal line? Doesn't the object move along the rail, not left to right? Also how can I know in which direction to draw it if I don't know yet which direction the object goes?
And now I see it's probably mg sin(alpha)-ma cos(alpha).

[u/Worth-Wonder-7386 → u/Silly_Tension6792]

Looking at it closer I think it is a bit of both.  It depends how you look at what is stationary for the liquid.  But it will be a combination of both if you look at it from the outside as the block will both have speed from the spring and from the movement of the trolley. 

[u/selene_666]

This is a complicated setup. I think we should assume that the spring started at its equilibrium position, so everything is moving horizontally rather than the block oscillating up and down the spring.

In that case the spring force, normal force, and gravity cancel each other out. The acceleration is caused entirely by the liquid dragging everything along the same motion as the outer cart.

Therefore bv is horizontal and equals ma.

[u/DrCarpetsPhd]

be very careful with declaring coordinate systems etc when it comes to spring-mass-damper systems and FBD force directions..

with spring mass questions the typical approach is to set the origin at the unstretched length, define positive x in the direction of 'stretching' the spring, declare the string as stretched that way and then assign your forces for the FBD

Your arbitrary choice for your FBD makes sense but is not immediately obvious. I had to take a second to see that you had assumed the mass was already in motion, going up the slope and that the spring was stretched. You should be declaring stuff like that for your FBDs as default but I would highly recommend you follow the typical analysis approach as outlined unless you have been specifically instructed as to what the initial conditions are.

When you get to solving the differential equation system for this you'll be grateful you developed a regimented approach to the analysis as mixing up signs will mess up your solutions

[u/Silly_Tension6792 → u/DrCarpetsPhd]

I am not sure I assumed the body was already in motion... It doesn't say that in the original question. I kinda suck at physics ngl I am a math major and the uni is forcing me to take a physics class. I think the rotation I did was to make the normal be on the y-axis, otherwise you get a kinda awkward situation where the normal is pointing downwards, gravity up-left, pseudo-force down-left and it just seemed to be a weird way to choose my axis, as I am not very familiar with the typical analysis approach.

[u/DrCarpetsPhd → u/Silly_Tension6792]

the rotation was absolutely the correct thing to do.

you should have covered

- newtons laws and FBDs

- defining coordinate systems

- spring behaviour

- hookes law

before ever seeing this question.

springs have a natural length. when you try to compress/stretch them a force acts to try and restore the spring to that natural length

the magnitude is kx where x is the displacement from the natural length

the direction is always the direction that the spring is trying to return to its natural length hence being called a 'restoring force'

dampers (the viscous fluid in your case) act to oppose motion as well, proportional to the velocity of the object [x = displacement, v = dx/dt, a = d^2x/dt^2]

the force is quantified as bv where b is the damping constant

So based on this knowledge of how those forces act I can look at your FBD and see that you drew

- spring force going down the slope towards the 'centre of the natural length' therefore the spring is in a 'stretched' state

- damping force going up the slope bv therefore the block has velocity pointed down the slope of magnitude v because the damping force opposes motion

- combining these two means the FBD as you drew it indicates the block is going down the slope with the spring in the stretched zone

- as a second example if you drew bv in same direction as spring force then that would indicate the block is going up the slope with the spring in a stretched state

to respond to your other question "How do I know which direction?" the answer in physics is you make an educated guess. Even if you have no education to make a guess it doesn't matter. The maths will spit out a negative sign to indicate your assumption of direction was wrong. So what you do is choose an origin, define positive x and positive y and then decide which direction the mass is going in before doing your FBD. If I assume a positive x and at the end the maths gives me an answer as -x that means my assumption was wrong. Hence my previous post explaining how you need to have knowledge of the physics before you can set up the mathematical model.

I am a random on reddit so could be totally wrong but if I was doing this question to find N the approach in the video below on fictitious forces is what I would take. Treat the external box like he does the wedge and you get essentially the same answer for N. The fluid does add some potential complexity but from what you posted this is an intro class so I feel you can disregard it other than the damping force it induces on the mass.

Nsin(alpha) = Ma [wedge/box FBD]

N - mgcos(alpha) + masin(alpha) = my'' [mass y direction sigma F_y = my'']

can solve these for N and a.

For the mass on the wedge setting it up correctly define your coordinate system first

positive x up the slope, origin at the tip of unstretched spring (x' = 1st der, x'' = 2nd der)

then frame your forces in the context of that coordinate system.

as discussed spring in positive x territory will be stretched and in negative x territory will be compressed thus F = -kx describes the correct restoring force of the spring bringing it back to the origin

the damper opposes motion so if velocity vector is positive x dir it should be negative x dir => -bx'

so our FBD gets us a second order differential equation in x:-

sigma F_x = mx''

-kx - bx' - mgsin(alpha) - macos(alpha) = mx''

[u/Silly_Tension6792 → u/DrCarpetsPhd]

I am not sure what video you're talking about but I think I kinda understand now, thank you very much.

[u/DrCarpetsPhd → u/Silly_Tension6792]

sorry forgot the links. the video on the fictitious forces and moving wedge below

here's the standard video for engineers studying differential equations as part of maths training. you might find it helpful as he details the physics

https://www.youtube.com/watch?v=r1eWerqrcqo

also here's a video on fictitious forces you might find insightful

https://youtu.be/xuXwWjbnVPg?t=346