Proof of T.a=0 rule in mechanics (Laws of motion)

In summary: Newton's equation gives the result you were looking for.In summary, the T.a rule can be applied to a system to solve problems regarding constraint motion. This example shows how the rule can be applied.
  • #1
Rinzler09
3
0
There is this T.a rule in laws of motion which can be applied to a system to solve problems regarding constraint motion. Here's an example
Physics Forums.jpg

This example is pretty simple so I've decided to show the application of the rule here.
Consider the FBD of m,
T is in the same direction as the acceleration. Therefore, T.a=Ta1
Considering the FBD of 2m,
T is in the opposite direction. Therefore, T.a=-Ta2

ΣT.a=0, Therefore, Ta1 - Ta2=0
Thus, a1=a2

This method is really useful for complex pulley systems such as this one.
pulleys.gif


I was wondering how to prove this. Can somebody help? Just give me an idea. Don't post the proof.
 
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  • #2
Well hello Rinzler, welcome to PF :smile: !

If you consider the constant length of the rope, it is pretty straightforward that y1 + y2 is a constant.
So v1 + v2 = 0 and a1 + a2 = 0 too.

The pulley system you draw is somewhat different, though
 
  • #3
BvU said:
Well hello Rinzler, welcome to PF :smile: !

If you consider the constant length of the rope, it is pretty straightforward that y1 + y2 is a constant.
So v1 + v2 = 0 and a1 + a2 = 0 too.

The pulley system you draw is somewhat different, though
Yeah, the length of the string is constant. But I'm saying that ΣT.a for the system is zero.
 
  • #4
Doesn't feel good to me: the dimension of ##\vec T \cdot \vec a## is all irregular.
##\sum \vec T \cdot \vec a = 0 ## only because ##\sum \vec a = \vec 0 ## and the T are equal.

Newton ##\sum \vec F = m\vec a## would be a lot better starting point for your analysis of e.g. the crate system.
And (with due care for the masses of the pulleys -- they can be different, equal, massless or all on one and the same axle) there will be an additional statement for the tensions.
 
  • #5
A single object accelerating under the action of a single tension (like a block pulled on a horizontal surface) does not satisfy this "rule".
As for the system in OP, why not sum of accelerations or sum of tensions? They are also zero but so what? It's not a general relationship.
 

1. What is the "Proof of T.a=0 rule" in mechanics?

The "Proof of T.a=0 rule" in mechanics is also known as the "proof of translational equilibrium". It states that if an object is in translational equilibrium, the sum of all the forces acting on it must be equal to zero. This means that the object is not accelerating and is either at rest or moving at a constant velocity.

2. What are the laws of motion that support this rule?

The laws of motion that support this rule are Newton's First Law, also known as the Law of Inertia, and Newton's Second Law, which describes the relationship between force, mass, and acceleration. These laws state that an object will remain at rest or in motion at a constant velocity unless acted upon by an external force, and that the net force on an object is equal to its mass multiplied by its acceleration.

3. How is this rule applied in real-world situations?

This rule can be applied in various real-world situations, such as the motion of objects on a flat surface, the motion of objects in fluids, and the motion of objects in space. It is also used in the design and analysis of structures, such as bridges and buildings, to ensure that they are in equilibrium and can withstand external forces without collapsing.

4. Can this rule be violated?

In theory, this rule cannot be violated. However, in practical situations, some external factors such as friction, air resistance, and imperfect measurements can cause the sum of forces to be slightly different from zero. This is why the rule is often referred to as the "proof" of T.a=0, as it is a simplified and idealized representation of real-world scenarios.

5. How does this rule relate to other principles in mechanics?

This rule is closely related to other principles in mechanics, such as the conservation of energy and momentum. In a system where T.a=0, the total energy and momentum of the system will also remain constant. This rule also supports the concept of equilibrium, which is essential in understanding the stability and motion of objects in various systems.

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