Why does the acceleration of a pendulum depend on Rθ and not x?

[aaaa202]

Okay I hope I can write this so it makes sense what I am thinking.
For a pendulum you have:

Fres = -mgsinθ
And Fres points along the circumference:
So:
mRθ'' = -mgsinθ

I wonna discuss this property that you can just express the acceleration as Rθ''. In a cartesian coordinate frame your acceleration would depend on the characteristic length scale of x. How do you know that the characteristic length scale of Rθ is the same as x? I know it sounds weird, but I hope you understand what I am going at.

 

[HallsofIvy]

I'm not sure what you mean by "characteristic length scale" but I suspect it has to do either with the units of measurement or a specific length in the problem.

The angle, [itex]\theta[/itex] is "dimensionless" so that "[itex]R\theta[/itex]" has the same units as R. Whatever "length scale" you are using for R also applies to [itex]R\theta[/itex].

 

[jedishrfu]

are you asking if X and Y are measured in meters then would r*theta be measured in meters?

 

[aaaa202]

no rather something like this: If we imagine that at some point the x-axis of our coordinate system tangential to the arc then the pendulum will move a distance dx measured in cartesian coordinates. How can we know that dx=rdθ, I mean what is it that says that these differentials are comparable? After all what if we made r bigger?

 

[Nickie]

The acceleration of a pendulum does not depend on the distance x because x is a linear measurement, while Rθ is a rotational measurement. The pendulum's motion is determined by the force of gravity acting on the mass of the pendulum, which creates a torque that causes the pendulum to swing back and forth. This torque is directly proportional to the distance R from the pivot point and the angle θ of the pendulum.

In other words, the acceleration of the pendulum is determined by the rotational motion of the pendulum, which is described by the variables R and θ. The distance x, on the other hand, is simply the linear displacement of the pendulum and does not take into account the rotational aspect of the pendulum's motion.

Furthermore, the characteristic length scale of Rθ is not necessarily the same as x. The characteristic length scale of Rθ is determined by the length of the pendulum arm and the angle of swing, while x is simply the distance from the pivot point to the mass of the pendulum. These two measurements are not equivalent and therefore cannot be used interchangeably when determining the acceleration of the pendulum.

In summary, the acceleration of a pendulum depends on Rθ because it takes into account the rotational motion of the pendulum, while x only measures the linear displacement. The characteristic length scales of Rθ and x are not the same and cannot be used interchangeably when calculating the acceleration of a pendulum.