# Energy and Power in Translating Mechanical Systems

## Introduction

This section discusses work, energy and power; three quantities that are of importance for translating physical systems.  The concepts are so useful that they are often applied to other types of systems.  For example, it is common to talk about a highly energetic stock market.  You should be familiar with these concepts from an introductory physics class.  They will only be briefly discussed here.

Contents

## Work

When a force is applied to an object, and that object moves, we can say that work has been done on the object.  For example, if a force is applied to a mass and the mass moves, work has been done.  However, if the mass does not move, no work has been done.  If a force is applied to a spring, and the spring compresses (or expands) work has been done to the spring.   If a dashpot has a force applied to it and it is compressed (or expanded), work has been done.   In the case of the mass and the spring, the work results in energy stored in the object; in the case of the dashpot, energy is dissipated and released to the environment as heat.

Mathematically work can be defined as the force applied multiplied by the distance moved.  For an object with a constant force, f, in the x direction, the work, W, is defined as

## Energy

Energy is the ability to do work.  For example, compressed spring stores energy that can be releases to do work on a physical system.  A battery stores energy that is released by a chemical reaction in the form of electrical energy (which can be converted to mechanical work by a motor).  In mechanical systems energy can be stored as potential or kinetic energy.

### Potential Energy

Potential energy can be stored in either masses or springs.  In a spring the potential energy, U, is given by

For a mass moved in a constant gravitation field

### Kinetic energy

Kinetic energy can only be stored in a mass.  The kinetic energy, T, is given by

Apply a change of variables and we get the expected result

### Dissipated Energy

When work is done to a dashpot, energy is not stored.  Instead heat is generated and lost to the environment.

We can do the same calculation for dry, or coulomb friction:

## Power

Power is a measure of the rate of change of energy in a system.

So if we push on a mass to accelerate it, the power required is

and this power goes into increasing the energy (i.e., velocity) of the mass.

If we push on a spring

and this power goes into increasing the energy (i.e., compression/elongation) of the spring.

If we push on a dashpot

and this power goes into creating heat, no energy is stored.  The dashpot simply dissipates energy.  Note that in all cases, the power is simply force multiplied by velocity.

Note: the previous page showed that power is conserved when there is a lever in the system, but there is a tradeoff between velocity and force.

##### Example: Sliding with coulomb friction

Consider a mass sliding with coulomb friction.  If it starts at t=0 with a velocity of vo, how far will it slide before stopping?

Original energy = ½mvo²
Dissipated energy=μkNx=μkmgx.

All energy is dissipated as heat so, μkmgx = ½mvo², or x=½vo²/(μkg)

##### Example: Maximum velocity

Consider the system shown.  If the initial position is xo and the initial velocity is zero, what will the maximum velocity be?

Since no energy is lost, the maximum kinetic energy (velocity) will occur at the point of minimum, or zero, potential energy (position).

½mvmax²  = ½kxmax²  = ½kxo² , so
vmax=xo(k/m)½

##### Example: Equations of motion

Consider the system shown.  Use the fact that energy is preserved to develop equations of motion

Since no energy is lost, the total energy is constant:

Since the energy is constant, the time derivative must be zero:

This is the same result we get from a free body diagram.

Note: this method is easily applied in situation such as this when there is no energy dissipated, but can be modified to take energy loss (friction) or gain (input sources).  Research the topic of "Lagrangian Mechanics," or "energy methods" if you are interested.

References