Question regarding setting up Newton's second law equations

[MrJoseBravo]

Homework Statement

So the question revolves around two blocks, one of mass A and one of mass B. The Mass A block is on a smooth horizontal surface, connected by a thin chord that passes over a pulley to the second block, of Mass B. The question asks to draw a free body diagram of both objects, and then apply Newtons second law to find the formulas for the acceleration of the system, and for the tension of the chord.

Homework Equations

block a: ƩF=Ft=ma

block b: ƩF=Ft-mg=ma

The Attempt at a Solution

I am able to draw the free body diagram, but when i try to find the formulas i get confused. I looked up the answer to the acceleration formula on the back of the book, and it is
g(m(b))/m(a)+m(b). I worked backwards to find this, but i had to re-arrange my block b equation to ƩF=mg-Ft=ma, in order to get the acceleration. What i did was i substituted the block B (Ft) with the block A (ma), and then solved for acceleration. My first question regards the block B equation, i could not solve for the answer with my original equation (Ft-mg) , i had to flip them. How do i know which force comes first in the equation? Secondly, was my substitution between the two formulas correct? Thank you for your help!

 

[PhanthomJay]

Two simultaneous equations can be solved by substitution or adding the equations together to elimimate one of the variables. Your substitution method is correct.

Note that acceleration is always in the direction of the net force. If the block on the table accelerates to the right, the hanging block must accelerate down. Therfore the net force on the hanging block must also act down. So which is greater..Mg or Ft?

 

[MrJoseBravo]

the Mg would be greater since it is accelerating downwards, right? I think i grasp it now, the acceleration depends on the net force, so the placement of the variables in the formula is determined by the motion of the object.

So the motion force comes first in the equations? is this a full-proof or are there instances where it could be different, say when there is a component force you have to take into consideration.

thanks for your help!

 

[PhanthomJay]

Acceleration is always in the direction of the net force which acts on an object. It is not always in the direction of the motion of the object. Suppose you throw a ball straight up in the air. It's motion is up, but the net force acting on it, it's weight, mg, acts down. So it's acceleration must be in the down direction, in the direction of the net force, not the motion. Or if you push a block up a plane with friction , then let go, both friction and the component of the gravity force act down the plane, so they add to give a net force down the plane, and the acceleration is down the plane. This latter example makes use of component forces ...it is often best to break up forces into their perpendicular component vectors, and look at accelerations in each of the directions independently.

 

[Nickie]

Hello,

It is great that you were able to draw the free body diagram for this problem. It is an important first step in solving any physics problem. Now, let's take a look at the formulas you have written down.

For block A, the equation you have written is correct: ƩF=Ft=ma. This means that the sum of all the forces acting on block A (which is only the tension force, Ft) is equal to the mass of block A times its acceleration.

For block B, the equation you have written is also correct: ƩF=Ft-mg=ma. This means that the sum of all the forces acting on block B (which is the tension force, Ft, and the weight force, mg) is equal to the mass of block B times its acceleration.

Now, let's take a look at your attempt at solving for the acceleration of the system. You have correctly set up the equations for both blocks, but you encountered some confusion when trying to solve for the acceleration of block B. This is completely normal, as it can be tricky to know which force to substitute in the equation.

In this case, the best approach would be to solve for the tension force, Ft, in the equation for block A (ƩF=Ft=ma) and then substitute that value into the equation for block B (ƩF=Ft-mg=ma). This way, you will have the same value for the tension force in both equations, which will make it easier to solve for the acceleration of the system.

To answer your first question, the order of the forces in the equation does not matter as long as you are consistent in your approach. In this case, you can choose to write the equation as ƩF=ma-Ft=mg or ƩF=mg-Ft=ma, as long as you are consistent throughout your calculations.

To answer your second question, your substitution between the two formulas was not entirely correct. When solving for the acceleration of the system, you cannot simply substitute one force for the other. Instead, you need to solve for the tension force and then substitute that value into the equation for block B.

I hope this helps clarify your confusion. Let me know if you have any further questions. Good luck with your homework!