Is this demonstration of the book from Ernst Mach correct?

In summary, The demonstration of the book from Ernst Mach is correct and based on the relationship between the radius and arc length, making it a rigorous and accurate explanation.
  • #1
jonjacson
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¿Is this demonstration of the book from Ernst Mach correct?

I am reading the book "The science of mechanics", Ernst mach.

He is talking about the achievements of Christian huygens, and he wants to calculate the centripetal acceleration of a circular motion.

As you can see in the image below :

o= Is the angle between the initial and final position of the particle

v= Is the initial speed

a= Is the acceleration

t= time

r= Is the radius of the circle

"On a movable object having the velocity v let a force act during the element of time t which imparts to the object perpendicularly to the direction of its motion the acceleration a. The new velocity component thus becomes a*t, and its composition with the first velocity produces a new direction of the motion, making the angle o with the original direction. From this results by conceiving the motion to take place in a circle of radius r, and on account of the smallness of the angular element, putting tan o=o, the following, as the complete expression for the centripetal acceleration of a uniform motion in a circle:

(From the triangle of the composition of speeds)

at/v=tan o

(From the triangle of the radius and the arc)

vt/r= o

If you aproximate the tangent for the angle, and this is correct for small angles you can arrive at:

a=v^2/r


But I have a question, I don't understand why the proportion of the speeds (v and at) is the same as the proportion of the distances in the circle (R to the arc lenght).

¿Could it be greater?¿Or smaller?¿WHy is the same?¿Is this demonstration completely rigorous?.
 

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  • #2
Yes, this demonstration from Ernst Mach is correct. The reason why the proportion of the speed and distance is the same is because it is based off of the relationship between the radius of the circle and the arc length. This means that the ratio of the speed and distance is equal to the ratio of the radius and the arc length. This makes the demonstration rigorous and accurate.
 

1. Is the demonstration of the book from Ernst Mach accurate?

The accuracy of Ernst Mach's demonstration in his book depends on several factors, including the experimental setup and the interpretation of the results. It is essential to carefully review and replicate the experiment to determine its accuracy.

2. How can we verify the correctness of Ernst Mach's demonstration?

The best way to verify the correctness of Ernst Mach's demonstration is to conduct independent experiments and compare the results. This process of replication and verification is an essential part of the scientific method.

3. What are the implications of Ernst Mach's demonstration?

Ernst Mach's demonstration has significant implications in the field of physics, particularly in our understanding of the relationship between mass and inertia. It also has implications for our understanding of physical laws and their applicability in different frames of reference.

4. Are there any limitations to Ernst Mach's demonstration?

Like any scientific experiment, Ernst Mach's demonstration also has its limitations. These limitations may include factors such as the precision of the equipment used, the control of variables, and the interpretation of results. It is essential to consider these limitations when evaluating the demonstration's validity.

5. How did Ernst Mach's demonstration contribute to the development of physics?

Ernst Mach's demonstration played a crucial role in the development of physics, particularly in the understanding of the principle of relativity and the relationship between mass and inertia. It also influenced the work of other scientists, such as Albert Einstein, who built upon Mach's ideas to develop the theory of general relativity.

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