How to Calculate Quantitative Energy Problems

The classical pendulum is one of the many problems in physics that is greatly simplified by employing conservation of energy.

In physics, the total energy of any closed system must remain constant regardless of any processes or interactions that occur within that system. This principle is known as conservation of energy, and it is one of the most fundamental assumptions about the natural world made by physics. As a result, many seemingly complicated problems in physics can be reduced to the comparatively simple task of balancing equations when tackled with a conservation of energy approach.

Instructions

1. Make sure the problem involves a closed system. If energy is escaping somehow (light radiating energy away or sound waves carrying energy away), then a conservation of energy approach will not be appropriate.

2. Calculate the total energy of the system at a certain point in time. Energy comes in many forms, but the following formulas represent a handful of common types of energy you may encounter.

   Kinetic energy of a body: KE = (1/2) x m x v^2, where KE is the kinetic energy of the body, m is its mass, and v is its velocity.

   Kinetic energy of a classical ideal gas at constant volume: U = c_v x n x R x T, where U is the kinetic energy of the classical ideal gas, c_v is the heat capacity, n is the amount of the gas in moles, R is the gas constant, and T is the temperature.

   Gravitational potential energy: U = m x g x h, where U is the gravitational potential energy of the body, m is its mass, g is acceleration due to gravity, and h is the height above a certain reference point.

   Rest mass energy: E = m x c^2, where E is the rest mass energy of the body, m is its mass, and c is the speed of light.

   Energy of a photon: E = h x nu, where h is Planck's Constant and nu is the frequency of the photon

   The total energy of the system will be the sum of all relevant energies present in the system.

3. Work backwards from your knowledge of the system's total energy to determine the specific energies at the desired point in time. For instance, if you know the total energy of a pendulum system and the gravitational potential energy of the pendulum at a specific point in time, you can then calculate the kinetic energy of the pendulum at that point in time because you know that kinetic energy has to make up the difference between the total energy and the gravitational potential energy.

4. Calculate specific variables as desired once you know what the different energies of the system at the desired point in time must be. Continuing with the pendulum example, once you know the kinetic energy of the pendulum at a given point in time, you could then calculate the pendulum's velocity at that point by rearranging the expression for kinetic energy to solve for velocity.

Tips & Warnings

• Any type of energy that you know will be invariant with respect to time in your system will cancel out and can be ignored. For example, a system will experience no change in rest mass energy unless very specific conditions involving relativity or nuclear physics are met. As a result, you can generally ignore rest mass energy in your calculations.

• The values of the different constants listed above vary based on what types of units you are approaching your problem in. Look up the constant values specific to your problem, and make sure all units agree before proceeding.

• The heat capacity, c_v, will be 3/2 for monatomic gases, 5/2 for diatomic gases, and 3 for larger molecular gases.

References

• Georgia State University: Hyperphysics