Can Resolving Forces Differently Lead to Contradictory Equations?

In summary, the friends encountered a mathematical paradox while solving a mechanics problem involving expressing T in terms of P. One friend used the "Human Method" and equated the vertical components, while the other friend resolved P instead. The question is whether T equals Pcosθ or Psecθ. However, the problem cannot be solved without additional constraint information, such as the absence of vertical acceleration. It is also important to consider the horizontal acceleration when attempting to solve this problem.
  • #1
HvB99
3
0
Hi ! So a friend and I were solving some mechanics problems in class today. And we came across a pretty funny mathematical paradox. So basically we tackled the problem in different ways...but we ended up with different equations...and none of us could prove the other wrong !
So here goes...
The Problem:
upload_2015-11-19_19-15-31.png

The question...a very simple one really :
"Express T in terms of P"
1) His method...(or as he describes it "The Human Method")
upload_2015-11-19_19-17-47.png

He resolved vertical component of the force T, ie. Tcosθ
Then he equated vertical components saying P = Tcosθ... therefore T = Psecθ

2)My way...
Note: This was part of a much complex problem... I'm not possessed to go through sooooo much trouble for such a small thing...and well...if i hadn't i wouldn't have found this...anyway
upload_2015-11-19_19-29-18.png

I resolved P instead,
and got that T = Pcosθ...the exact opposite

SO...the big question is ...
T = Pcosθ vs. T = Psecθ

Now ... i know practically speaking I might be wrong ...because according to me T < P ...however T would have to be greater ...since its offsetting the downward vertical force of P and has a leftward horizontal component...but then again ...can't you make the same argument about resolving P ??

I know .. I've been blabbering alot...but it really seems mind boggling !
Mathematically both seem correct XDUpdate : There isn't any vertical acceleration !

-HvB99
 
Last edited:
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  • #2
You cannot express P in terms of T without some additional constraint information. For example, the condition that there is no vertical acceleration or something similar.
 
  • #3
there is no paradox. you can't solve this problem with this method because T and P is not in same line. one vector can't generate a plane.the first one you don't consider the vertical force of T and the second one you don't consider sinx of P. if you want to express vectors, you need 2 vectors to describe the other one.
 
  • #4
DaleSpam said:
You cannot express P in terms of T without some additional constraint information. For example, the condition that there is no vertical acceleration or something similar.

Yes...there isn't ...this was known...but i forgot to mention it...sorry!
 
  • #5
hesher said:
there is no paradox. you can't solve this problem with this method because T and P is not in same line. one vector can't generate a plane.the first one you don't consider the vertical force of T and the second one you don't consider sinx of P. if you want to express vectors, you need 2 vectors to describe the other one.

I'm sorry... but the other component...the one that isn't accounted for...as you said ...is the one causing a horizontal acceleration, since it is unbalanced( only that component ) ...

I merely equated the components of the two forces that were in the same direction.
 
  • #6
How about the horizontal acceleration? What do you know about it?

By the way, with just these two forces you cannot have and equilibrium.
And it is good practice to state the problem completely before jumping to (or expecting) a "solution".
 
  • #7
HvB99 said:
however T would have to be greater ...since its offsetting the downward vertical force of P and has a leftward horizontal component...
Given the additional information on the acceleration this is exactly correct. The vertical component of T is equal to P, and additionally T has a horizontal component that P does not.
 

Related to Can Resolving Forces Differently Lead to Contradictory Equations?

What is the equilibrium of forces paradox?

The equilibrium of forces paradox is a concept in physics that refers to a situation where the forces acting on an object are balanced, resulting in a state of equilibrium. This means that the net force on the object is equal to zero, and the object will remain at rest or continue to move at a constant velocity.

How is the equilibrium of forces paradox related to Newton's first law of motion?

The equilibrium of forces paradox is closely related to Newton's first law of motion, also known as the law of inertia. This law states that an object at rest will remain at rest and an object in motion will continue in motion at a constant velocity unless acted upon by an unbalanced force. In the case of the equilibrium of forces paradox, the object is at rest or moving at a constant velocity due to the balanced forces acting on it.

Can the equilibrium of forces paradox be applied to real-life situations?

Yes, the equilibrium of forces paradox can be applied to many real-life situations. For example, a book sitting on a table is in a state of equilibrium because the force of gravity pulling the book down is balanced by the normal force of the table pushing up on the book. Similarly, a car traveling at a constant speed on a flat road is also in a state of equilibrium, as the forces of friction and air resistance are balanced by the engine's force pushing the car forward.

What happens to the equilibrium of forces when an external force is applied?

When an external force is applied to an object in a state of equilibrium, the forces are no longer balanced, and the object will either start to move or change its velocity. For example, if you push a book sitting on a table, the forces will become unbalanced, and the book will begin to move.

How can the equilibrium of forces be calculated and represented mathematically?

The equilibrium of forces can be calculated using vector addition, where all the forces acting on an object are represented as vectors and added together. If the net force is equal to zero, the object is in equilibrium. Mathematically, this can be represented as ΣF = 0, where ΣF is the sum of all the forces acting on the object and must equal zero for equilibrium to be achieved.

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