Study Notes on Elastic and Electric Potential Energy

Definition of Elastic Potential Energy: This is the energy stored by stretching or compressing materials.
Hooke's Law: This law states that the extension or compression of an object is proportional to the force applied to it. * Proposed by: Robert Hooke, who is also historically recognized for discovering cells.
Spring Constant ( $k$ ): * Defined as the force of an object divided by the distance it is stretched. * Formula: $k = \frac{f}{d}$ . * Measurement Unit: Measured in Newton meters ( $N/m$ ).
Formula for Elastic Potential Energy ( $U$ ): * $U = \frac{1}{2} k \cdot d^2$ * Components: * $U$ : Potential Energy (PE). * $k$ : Spring constant. * $d$ : Displacement distance.

Problem Statement: A spring is stretched by $1\,m$ using $2\,N$ of force. What is the potential energy of the spring? Show your work and draw a model.
Given Variables: * Distance ( $d$ ) = $1\,m$ * Force ( $F$ ) = $2\,N$
Step 1: Calculate the Spring Constant ( $k$ ): * Using $k = \frac{F}{d}$ * $k = \frac{2\,N}{1\,m}$ * $k = 2\,N/m$
Step 2: Calculate Potential Energy ( $U$ ): * Using $U = \frac{1}{2} k \cdot d^2$ * $U = \frac{1}{2} \times (2\,N/m) \times (1\,m)^2$ * $U = 1\,\text{joules}$
Model States: * Static: The resting state of the spring. * Stretched: The state of the spring under tension. * Compressed: The state of the spring when pushed inward.

Definition: Electric Potential Energy is used to calculate the potential energy existing between two charges.
Conceptual Analogy: It is compared to the energy involved in holding two magnets apart.
Electric Potential Energy Formula: * $U = k \frac{q_1 \cdot q_2}{d}$ * Variables: * $U$ : Potential Energy (PE). * $k$ : Coulomb's Constant (approximately $9 \times 10^9$ ). * $q_1$ : Charge 1. * $q_2$ : Charge 2. * $d$ : Distance or separation between the charges.

Problem Statement: A proton is $2\,nm$ (nanometers) from an electron. If the charge of the proton is $1.00\,nC$ (Nano-Coulombs) and the electron is $-1.00\,nC$ , what is the potential energy holding them apart? Show your work and draw a model.
Given Variables: * Distance ( $d$ ) = $2\,nm$ ( $2 \times 10^{-9}\,m$ ). * Proton Charge ( $q_1$ ) = $1.00\,nC$ . * Electron Charge ( $q_2$ ) = $-1.00\,nC$ . * Coulomb Constant ( $k$ ) = $9 \times 10^9$ .
Calculation and Substitution: * The transcript provides the calculation: $V = k \frac{\text{charge 1} \cdot \text{charge 2}}{d}$ * $U = \frac{9 \times 10^9 \cdot (1) \cdot (-1)}{2\,nm}$
Result: The potential energy is stated as $4500\,\text{joules}$ .