Let's say I am trying to derive the equation for the hanging cable. The solution is a catenary curve. My question is:
In the setup for this problem, one would begin by considering some small segment of the string and analyzing the forces on it. Because it is in static equilibrium, the forces must all sum to zero. Because of the symmetry of the problem, we will consider the origin (0,0) to be the midway point where the chain is the lowest. Then we want to find y=ƒ(x); in other words, some equation describing the height of the chain as a function of x. Let us call the mass per unit length μ, the tension force T, the force due to gravitation g, and Δx the length of the string segment.
The gravitational force would act directly downward, and the tension force would act in two different directions at different angles. Here would be the equations describing the force
x dimension:
-Tcos(Θ) +Tcos(Θ+ΔΘ)=0
y dimension:
-μ(Δx)g - Tsin(Θ) +Tsin(Θ+ΔΘ)=0
However, at this point I realize that I must be in error. If we assume that the tension force is the same (and that T≠0) throughout the string, then in the 'x' dimension it will be impossible for the forces to sum to zero. Because in general cos(Θ)≠cos(Θ+ΔΘ), we have a contradiction:
-Tcos(Θ) +Tcos(Θ+ΔΘ)=0
T(-cos(Θ) +cos(Θ+ΔΘ))=0
T=0
or
-cos(Θ) +cos(Θ+ΔΘ)=0
cos(Θ)=cos(Θ+ΔΘ)
So what assumption is wrong here? Thanks for any help.
In the setup for this problem, one would begin by considering some small segment of the string and analyzing the forces on it. Because it is in static equilibrium, the forces must all sum to zero. Because of the symmetry of the problem, we will consider the origin (0,0) to be the midway point where the chain is the lowest. Then we want to find y=ƒ(x); in other words, some equation describing the height of the chain as a function of x. Let us call the mass per unit length μ, the tension force T, the force due to gravitation g, and Δx the length of the string segment.
The gravitational force would act directly downward, and the tension force would act in two different directions at different angles. Here would be the equations describing the force
x dimension:
-Tcos(Θ) +Tcos(Θ+ΔΘ)=0
y dimension:
-μ(Δx)g - Tsin(Θ) +Tsin(Θ+ΔΘ)=0
However, at this point I realize that I must be in error. If we assume that the tension force is the same (and that T≠0) throughout the string, then in the 'x' dimension it will be impossible for the forces to sum to zero. Because in general cos(Θ)≠cos(Θ+ΔΘ), we have a contradiction:
-Tcos(Θ) +Tcos(Θ+ΔΘ)=0
T(-cos(Θ) +cos(Θ+ΔΘ))=0
T=0
or
-cos(Θ) +cos(Θ+ΔΘ)=0
cos(Θ)=cos(Θ+ΔΘ)
So what assumption is wrong here? Thanks for any help.