Normal Forces on a Sphere in a Non-vertical Groove

[Quasar100]

Homework Statement

Different Solving Way

Relevant Equations

Lami Theorem(?)

neglect friction and motion (sliding) and G(sphere)=20N. In this question I reached two different result with two different solving method.But one of them is false according to answer key. My question is why first solving way is false? Because the first solution way makes sense to me. If we divide the vectors into parts, isn't the force coming in this direction absorbed?(ex: reaction force Nz parallel to the x surface)

So that means that this solving way is true.
The second way is gathering the vectors of forces in one center(in situation of balance) and using lami theorem.

 

[BvU]

Hello @Quasar100 ,

$\qquad$ !​

In this question

What question ?

Please read https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/

$\$

 

[haruspex]

why first solving way is false?

I can't see an actual "way". Just a lot of numbers and labels linked by arrows.
I have no idea what your method is.
Describe what you are doing (like, "balance of moments about point …") and write equations.

 

[willem2]

Homework Statement:: Different Solving Way
Relevant Equations:: Lami Theorem(?)

eglect friction and motion (sliding) and G(sphere)=20N. In this question I reached two different result with two different solving method.But one of them is false according to answer key. My question is why first solving way is false? Because the first solution way makes sense to me. If we divide the vectors into parts, isn't the force coming in this direction absorbed?(ex: reaction force Nz parallel to the x surface)

It seems like the first line of the question has disappeared somewhere. I think I can still make out the question.
You have 3 forces (gravity, the normal force from two planes) that must sum to 0, because there is an equilibrium. The normal forces have known direction, but unknown magnitude. Gravity is 20 Newton and straight down. The question asks for the size of the normal forces.

Yourr first solving method might possibly equate horizontal and vertical components of the forces, wich should work. The way you have written down the answer is however so unclear, that I can't tell if that's really what you are trying to do, or where any mistake might be.

 

[Quasar100]

Hello @Quasar100 ,

$\qquad$ !​

What question ?

Please read https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/

$\$

sorry I forgot it. The question is: What is reaction force of Z surface to the sphere?

 

[Quasar100]

Yourr first solving method might possibly equate horizontal and vertical components of the forces, wich should work. The way you have written down the answer is however so unclear, that I can't tell if that's really what you are trying to do, or where any mistake might be.

Yes, what I've done is this: Dividing the vectors into their x and y components and thus finding the reaction force of the Z surface.

 

[Quasar100]

It seems like the first line of the question has disappeared somewhere. I think I can still make out the question.
You have 3 forces (gravity, the normal force from two planes) that must sum to 0, because there is an equilibrium. The normal forces have known direction, but unknown magnitude. Gravity is 20 Newton and straight down. The question asks for the size of the normal forces.

Yourr first solving method might possibly equate horizontal and vertical components of the forces, wich should work. The way you have written down the answer is however so unclear, that I can't tell if that's really what you are trying to do, or where any mistake might be.

In the second picture, the first calculation(yellow writings), what I did was to find the reaction force of the X surface. Second calculation(green writings) was to find the reaction force of the Z surface!

 

[Quasar100]

View attachment 322886View attachment 322887
Question: Neglect friction and motion (sliding) and G(sphere)=20N. What is the reaction force exerted by the surface Z on the sphere?. In this question I reached two different result with two different solving method. But one of them is false according to answer key. My question is why first solving way is false? Because the first solution way makes sense to me.
First Method: If we divide the vectors into parts(x and y components), isn't the force coming in this direction(below picture) absorbed?(because as an example x component of Nx(x surface reaction force to the sphere) ) will be absorbed. It is because parallel with Z surface. Same situation exist for reaction force of Z surface. View attachment 322888
I mean small portion of the weight force will be wasted.
So that means that this solving way is true.
The second way is gathering the vectors of forces in one center(in situation of balance) and using lami theorem. But in this way reached a different result. View attachment 322899 In here, the angle between the reaction forces was tried to be found. The final version is:View attachment 322900
Here we find the answer as 20N. Which is also answer key answer. But my question is Why the first solution is wrong. Whereas, when we use the lami theorem, we don't take absorption into account.
Sorry, I am begginer. So Since I couldn't take the 360 minutes editing timeout into account, I reorganized and rewrote it here.

 

[haruspex]

The use of X, Y and Z to label the planes is unfortunate, given that you also want to use x and y for horizontal and vertical components. For clarity, I suggest being strict about using uppercase for the first and lowercase for the second.

It is because parallel with Z surface.

If you mean that the horizontal component of N_X will be equal and opposite to the horizontal component of N_Z, yes. Otherwise, what do you mean?

small portion of the weight force will be wasted.

No idea what you mean by that.

So that means that this solving way is true.

What way? As I asked before, please write equations.

 

[Quasar100]

If you mean that the horizontal component of NX will be equal and opposite to the horizontal component of NZ, yes. Otherwise, what do you mean?

No, That's not what I mean. Let me try to explain this through an example. A force of 20 N is applied to a square object. The question is what is the reaction force of the surface? (In the below picture).

We found N_surface as 12 N. Because we did not include the x component of force in the calculation of the reaction force. Because the x component of the force and the surface are parallel(That's what I mean by waste). The question I asked above has similar situation(If we try to solve it by separating it into its x and y components as in the example I gave). Imagine that the object(square object) is not moving. It is futile to apply force parallel to the x-axis. In the question above, the sphere does not move because it is in equilibrium. Therefore, the reaction force of Z is perpendicular to the Z surface but not exactly perpendicular to the X surface. What I mean is that the x component of the reaction force on the X surface will be parallel to the X surface. For this reason, the weight force(G) decreased 20 N to 10√3 as it transforms into the reaction force of the X surface. Again, the reaction force of the X surface decreased as it transforms into the reaction force of the Z surface and became 15N. But this answer is wrong. That's exactly my question: Why is it wrong? What part of the question am I wrong? Because I reach the correct answer with the second solution way(with lami theorem). But in the lami theorem, there is no such thing as the waste of force that I want to talk about in this post.

 

[Quasar100]

View attachment 322886View attachment 322887
Question: Neglect friction and motion (sliding) and G(sphere)=20N. What is the reaction force exerted by the surface Z on the sphere?. In this question I reached two different result with two different solving method. But one of them is false according to answer key. My question is why first solving way is false? Because the first solution way makes sense to me.
First Method: If we divide the vectors into parts(x and y components), isn't the force coming in this direction(below picture) absorbed?(because as an example x component of Nx(x surface reaction force to the sphere) ) will be absorbed. It is because parallel with Z surface. Same situation exist for reaction force of Z surface. View attachment 322888
I mean small portion of the weight force will be wasted.
So that means that this solving way is true.
The second way is gathering the vectors of forces in one center(in situation of balance) and using lami theorem. But in this way reached a different result. View attachment 322899 In here, the angle between the reaction forces was tried to be found. The final version is:View attachment 322900
Here we find the answer as 20N. Which is also answer key answer. But my question is Why the first solution is wrong. Whereas, when we use the lami theorem, we don't take absorption into account.
Sorry, I am begginer. So Since I couldn't take the 360 minutes editing timeout into account, I reorganized and rewrote it here.

The last 2 pictures don't show up for me, so I'm uploading them here again.(My solution with the lami theorem.)

N_Z is found to be 20 from this equation

 

[haruspex]

We found Nsurface as 12 N. Because we did not include the x component of force in the calculation of the reaction force. Because the x component of the force and the surface are parallel(That's what I mean by waste).

Ok, but that is just saying N is equal and opposite to the component of F that is normal to the surface.

Imagine that the object(square object) is not moving. It is futile to apply force parallel to the x-axis.

I don't know what you mean by that. Whether it is futile depends on what you are trying to achieve.

the x component of the reaction force on the X surface will be parallel to the X surface.

You haven’t defined your axes, so it is unclear what you mean by x component. You seem to be taking it to be parallel to whatever surface is currently under consideration.

the weight force(G) decreased 20 N to 10√3 as it transforms into the reaction force of the X surface.

Yes, that is the component of G that is normal to the X plane. But there is also a component of the normal force from the Z plane, transmitted through the sphere, so the normal force from X will be rather more.

the reaction force of the X surface decreased as it transforms into the reaction force of the Z surface and became 15N.

But you have not yet found the reaction force from X.
Because of the interaction between the two normal forces, you cannot solve it sequentially. You need to write a couple of simultaneous equations.

 

[Quasar100]

You haven’t defined your axes, so it is unclear what you mean by x component. You seem to be taking it to be parallel to whatever surface is currently under consideration.

Yes.

Yes, that is the component of G that is normal to the X plane. But there is also a component of the normal force from the Z plane, transmitted through the sphere, so the normal force from X will be rather more.

I understood that too.

But you have not yet found the reaction force from X.
Because of the interaction between the two normal forces, you cannot solve it sequentially. You need to write a couple of simultaneous equations.

Now I get it. I took it one by one(sequantially). But since it was in equilibrium, it must have been calculated simultaneously, as you said.
Many thanks @haruspex