Newton's Second Law and Inertial Frames Of Refence

In summary, Newton's Second Law states that the net force acting on a body is equal to the time rate of change of the body's linear momentum. This law is only applicable in an inertial frame of reference, where the last term in the equation is zero. However, in an accelerating frame of reference, this term may not be zero and can be interpreted as a fictitious force. Further testing in non-inertial frames is necessary to confirm this.
  • #1
cwstrick
1
0
I have a question about Newton's Second Law and Inertial Frames of Refrence. It is canon that Newton's Second Law is only applicable in an inertial frame of refrence. Newton's Second Law is the net force acting on a body is equal to the time rate of change of the body's linear momentum. Expressed mathematically N.S.L is

[tex]\sum[/tex] F = dp/dt = d(mv)/dt = (dm/dt)v + m(dv/dt)

Lets define a general velocity vector using arbitrary coordinates.The Einstein Summation Convention is being used. Let v = vaea
then (dv/dt) = (dva/dt)ea + va(dea/dt). Substituting this into the equation above and factoring one can arrive at

[tex]\sum[/tex] F = [(dm/dt)va + m(dva/dt)]ea + mva(dea/dt).

In regular cartesian coordinates, the last term in zero. My question is, can one apply N.S.L. in an accelerating frame of refrence by adding the last term, mva(dea/dt), instead of adding ficticious forces?
 
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  • #2
That last term probably is the fictitious force. Try it with a standard non-inertial frame like a rotating or uniformly accelerating system and see.
 
  • #3


Thank you for your question. I can provide a response to your query about Newton's Second Law and Inertial Frames of Reference.

Firstly, it is important to understand that Newton's Second Law is a fundamental principle in classical mechanics and it states that the net force acting on a body is equal to the time rate of change of the body's linear momentum. This law is valid only in inertial frames of reference, which are frames that are not accelerating or rotating. In these frames, the laws of physics are the same for all observers and there is no need to introduce any additional forces.

Now, let's consider your question about applying Newton's Second Law in an accelerating frame of reference. In such a frame, the laws of physics are not the same for all observers and we need to introduce fictitious forces to account for the acceleration. These forces are not real, but they are necessary to explain the motion of objects in the accelerating frame.

In your equation, you have correctly factored in the fictitious forces by including the term mva(dea/dt). This term accounts for the acceleration of the frame and it is necessary to obtain the correct result when applying Newton's Second Law. Therefore, in an accelerating frame of reference, you cannot simply ignore this term and apply the law as if the frame were inertial.

In conclusion, Newton's Second Law is only valid in inertial frames of reference and in an accelerating frame, we need to include fictitious forces to correctly apply this law. I hope this explanation helps to clarify your question.
 

1. What is Newton's Second Law?

Newton's Second Law states that the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass.

2. Can you explain the concept of an inertial frame of reference?

An inertial frame of reference is a coordinate system in which Newton's laws of motion hold true. This means that an object at rest will remain at rest and an object in motion will continue to move at a constant velocity unless acted upon by an external force. It is a frame of reference that is not accelerating or rotating.

3. How does Newton's Second Law apply to inertial frames of reference?

In an inertial frame of reference, Newton's Second Law serves as a basis for understanding the relationship between force, mass, and acceleration. It states that if the net force on an object is zero, the object will have zero acceleration and will remain at rest or continue moving at a constant velocity.

4. Can you give an example of Newton's Second Law in action in an inertial frame of reference?

An example of Newton's Second Law in an inertial frame of reference is a car traveling at a constant speed on a straight road. In this scenario, the car's acceleration is zero because there is no net force acting on it. If the driver suddenly applies the brakes, the car will decelerate due to the force of friction between the tires and the road.

5. What is the significance of Newton's Second Law in understanding the behavior of objects in motion?

Newton's Second Law is essential in understanding the relationship between force, mass, and acceleration, and how these factors affect the motion of objects. It allows us to make predictions about the behavior of objects in motion and is the foundation of classical mechanics.

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