Contents

Potential Energy¶

It is kinetic energy that is stored in an object. (Associated with the objects weight and position above ground).

Forces associated with potential energy are called conservative forces (The total mechanical, kinetic, potential is constant => conservative system).

We can view potential energy as “potential” for a force to do work. As we raise a pile driver hammer we give it potential to do work as the hammer falls.

As kinetic energy increases potential energy decreases.

Gravitational potential energy¶

Here we interested of the work done by the weight when it drops from height \(y_i\) to a smaller height \(y_f\).

\[\begin{split} W_{\text{grav}} = Fs \\ = mg(y_i - y_f) = mgy_i - mg y_f \\ = U_{\text{grav}, i} - U_{\text{grav}, f} = - \Delta U_{\text{grav}} \end{split}\]

Thus as an object falls its potential energy decreases. If we move an object upwards \(W_{\text{grav}}\) grows (If it would fall it would generate more energy). The potential gravitational energy is defined as:

\[ U_{\text{grav}} = mgy \]

  • \(y\) is the height above the origin

There is a important relationship between kinetic and potential energy described by total mechanical energy.

Elastic potential energy¶

Here we relate the work done by compression (expansion) a string with potential energy.

\[ W_{\text{el}} = \frac{1}{2}kx_i^2 - \frac{1}{2}kx_f^2 \]

  • \(x_f \ge x_i \Rightarrow W_{\text{el}} < 0\) , thus an object moves in \(+x\) direction while the string pulls in \(-x\) direction

The string potential energy is defined as:

\[ U_{el} = \frac{1}{2}kx^2 \]

  • units of \(k = N/m\)

  • x is the displacement of the string from its upstretched length

We can redefine work as:

\[ W_{\text{el}} = \frac{1}{2}kx_i^2 - \frac{1}{2}kx_f^2 = U_{text{el},i} - U_{text{el},f} \]

  • as x increases \(W_{\text{el}}\) is negative and the stored energy \(U_{\text{el,i}}\) increases

  • as x decreases \(W_{\text{el}}\) is positive and the stored energy \(U_{\text{el,i}}\) decreases

Markov chain Monte Carlo (MCMC) inference Bayes rule