# Finding the tangential force experienced by a bob of mass m on a simple pendulum via the gradient/nabla operator)

The problem was posed as follows. Given a pendulum of length $L$ with a mass $m$ attached to it, which forms an angle $\theta$ from the y-axis to the direction of swinging.

First we had to find the potential energy as a function of the angle, which was trivial enough.

$$U = mgL(1 - cos(\theta))$$

But the next question was to develop the gradient for this potential energy function via the arc length $s = L\cdot\theta$, where $s$ is the arc length.

I calculated the gradient by using the trigonometric definition of $\cos(\theta)$, but I developed the gradient by changing the variable of my function $U(\theta)$ to $U(x,y)$ after which I calculated it further, and found that $F_{tan} = -mg\sin(\theta)$, as desired.

But is there an alternative way to calculate it, by using the gradient, and by utilizing the arc length formula above? The question explicitly asks to develop the gradient via the arc length rather than $x$ and $y$ components.

• Please try to use math formatting for better readability. – ja72 Oct 4 '13 at 16:24
• Do you know about the gradient in polar coordinates? – ja72 Oct 4 '13 at 16:26
• I am not familiar with it, but after looking at some pages online, I am not quite sure how to proceed. In polar coordinates would my equation be U(L, a)? Doesn't this pose a problem as the radius is constant, and if I understood what I am read, then the entire gradient would reduce to (1/L)*(partial derivative of U to a), which would just give me (1/L)(mgLsin(a)), which leaves me with my original equation mgsin(a), which still does not include the arc length. – Kayle of the Creeks Oct 4 '13 at 17:54

Writing $U$ as a function of $x$ and $y$ doesn't really make sense because the length of the pendulum is fixed at $L$ so that the system is really one dimensional.
I think what you were supposed to do is change variables from $\theta$ to $s$. Notice $\theta=s / L$. Plugging this into the expression for $U$, we get $$U(s) = mgL(1- \cos(s/L)).$$ Next I think you are supposed to calculate the derivative of the energy in this coordinate system. You should get $$\frac{d}{ds} U = mg \sin(s/L).$$