Acceleration of Mass m: Problem Solving

[pizzamakeren]

Homework Statement

I need help solving a problem related to acceleration.

Relevant Equations

The relevant equation will be postet as a picture below

The problem that i am facing has to do with acceleration. The problem states that we have two balls with the same mass m that stands on two different tables. Between these balls are a wire with the length 2l, which i assume means 2 * l, where l is a value i choose. In the middle of the wire (x = 0) a force F pulls down the rope.

The problem asks me to show that the following equation gives me the acceleration of m. It is also noted that we are going to ignore friction.
Now i am wondering if the solution to this is to plug in values for F, m and l that i choose for myself? Or is it more complicated than that?

 

[Lnewqban]

The force pulling the balls together will not be constant; therefore, acceleration will vary as well.

 

[kuruman]

Homework Statement:: I need help solving a problem related to acceleration.
Relevant Equations:: The relevant equation will be postet as a picture below

The problem that i am facing has to do with acceleration. The problem states that we have two balls with the same mass m that stands on two different tables. Between these balls are a wire with the length 2l, which i assume means 2 * l, where l is a value i choose. In the middle of the wire (x = 0) a force F pulls down the rope.

The problem asks me to show that the following equation gives me the acceleration of m. It is also noted that we are going to ignore friction.
Now i am wondering if the solution to this is to plug in values for F, m and l that i choose for myself? Or is it more complicated than that?
View attachment 298453

No. You are not asked to substitute values in a formula to find the acceleration. You are asked to prove that the formula as given to you is correct. Let's say you measured the acceleration somehow and got a certain value, say 2.3 m/s². Then let's say you measured F, m, x and l and you combined them as $\dfrac{F}{2m}\dfrac{x}{\sqrt{l^2-x^2}}$ would you get 2.3 m/s² or not? What makes you so sure that this combination will give you 2.3 m/s² as opposed to $\dfrac{2F}{m}\dfrac{x}{\sqrt{l^2-x^2}}$?

You need to start with Newton's second law and derive an expression for the acceleration of each mass.

It would also help us help you if you provided the statement of the problem exactly as given to you.

 

[pizzamakeren]

The statement of the problem is as accurate as it can get. I've translated it from norwegian.
Newtons second law says that F = m * a, which means i can get a equation for a. a = F / m
What do you mean for each mass? The masses of the balls are the same, or am i misunderstanding something? I haven't gotten any value for the acceleration.

 

[haruspex]

What do you mean for each mass? The masses of the balls are the same

The word mass is a little ambiguous in English. It can refer to a particular magnitude of mass, or to a particular instance of mass - as in "left hand mass", "right hand mass".
Have you drawn a free body diagram for one mass? What equations can you write from it?

 

[kuruman]

Each mass has its own acceleration. In this case the acceleration of one mass has the same magnitude but opposite direction to other mass's acceleration. You are supposed to find an expression for that magnitude. The problem gives you that expression to guide your thinking. For this particular situation, you need to start with F_net = ma and end up with $\dfrac{F}{2m}\dfrac{x}{\sqrt{l^2-x^2}}$ not with $a=F/m$. That's because the net force on a mass, as @Lnewqban already noted, is not constant but depends on $x$. To find the net force you need to draw a force diagram or free body diagram (FBD). Have you learned how to do this?

I see @haruspex responded with a similar message, but I will leave mine as is.

 

[pizzamakeren]

unfortunately i haven't learned how to do this. I better check out how to do this.

 

[kuruman]

unfortunately i haven't learned how to do this. I better check out how to do this.

I think that would be a good idea in general. For this problem in particular, you need to find the component of the net force in the direction of the acceleration. When the middle of connecting wire is at distance $x$ from it initial horizontal position, the force pulling on a mass is down and across but the acceleration is directed only across because the table prevent it from moving down.

 

[pizzamakeren]

I've rewritten the problem so that it might be easier to understand:
Two balls lie on a table, each ball has a mass of m, and they are connected to each other
with a cord that has a length of 2l. A constant force ~ F pulls in the midpoint
on the string (x = 0), normally on the string's original position. Show that the acceleration of m in the
normal direction of ~ F is given by:

This is the drawing i got for the problem:

I don't know if this changes anything, but ill try looking more into it.
Im not sure about how i can turn F = ma into:

Is there a special method i need to use?

 

[PeroK]

I've rewritten the problem so that it might be easier to understand:
Two balls lie on a table, each ball has a mass of m, and they are connected to each other
with a cord that has a length of 2l. A constant force ~ F pulls in the midpoint
on the string (x = 0), normally on the string's original position. Show that the acceleration of m in the
normal direction of ~ F is given by:
View attachment 298506

This is the drawing i got for the problem:
View attachment 298507
I don't know if this changes anything, but ill try looking more into it.
Im not sure about how i can turn F = ma into:
View attachment 298508Is there a special method i need to use?

Have you thought about using trigonometry?

 

[haruspex]

This is the drawing i got for the problem:
View attachment 298507

In your own diagram, you will need to show a later position, where x>0.