- #1
jonjacson
- 447
- 38
¿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?.
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?.