# Force, Energy, Work, and Momentum

Force, Energy, Work, and Momentum

Law of the conservation of mechanical energy

Clinical applications of Mechanical Energy

## Kinematics

This is about the description of motion as a function of time. Problems will often ask you to connect the concepts discussed here to the kinematics equations to calculate answers. See the page on the Kinematics Equations for specifics.

## Newton’s Laws

### 1st Law

An object in motion will stay in motion, and an object at rest will stay at rest (Surprising to us because we assume friction is everywhere, so we intuivitely know the second part). Think about the voyager satellite -- Having left our solar system, there is nothing with which it can interact, so it will move in the direction it is heading at its current speed for millions of years (the next solar system it runs into.)

### 2nd Law

Force = how hard you push or pull on an object. Acceleration = change in velocity. Mass = the amount of actual matter that you have. This is the quantity that resists a change in velocity (in other words, mass is what has inertia, and thus, the basis of the First Law).

Newton’s Second Law:

Force = mass × acceleration

F = ma

Note:

- Newton’s 2nd law tells us about acceleration, not motion. If something is moving, it has a velocity that is nonzero.
- 1st Law can be seen as special case of 2nd (Fnet = 0 because no F acting)

### 3rd Law

For every action, there is an equal and opposite reaction. The reaction force is equal in type AND magnitude, opposite in direction. By type, I mean that if the action force is contact, electrostatic, magnetic, or gravitational, the reaction force is the same. Other forces may be the ones arresting motion (like the Normal Force vs. gravity), but that is not the subject of the Third Law. Some examples:

Two skaters on ice, pushing against each other. The force is equal, the smaller person (the lady), has less mass, so she accelerates faster:

You fall out of a boat that is floating on a lake (frictionless surface). As you fall forward, the boat accelerates in the opposite direction:

You are standing on the Earth, being attracted to it by gravity. You don’t sink in because of the Normal Force (see below) exerted by the ground on the soles of your feet. However, the Normal Force IS NOT the reaction force to your weight. That is the gravity that you exert on the Earth, pulling it up. The normal force is a version the Electrostatic force (see below).

## Normal Force

Caused the repulsion of the negatively charged electrons clouds on the surfaces of the two solid objects being pressed together. Not always equal to gravity! Equal in magnitude to the forces causing the objects to be smushed together, oriented normal (perpendicular) to the surfaces. The situation on the right generates more normal force, in a direction opposite to the arrow drawn (which is the direction of smushing).

## Friction

fs ≤ μsFn (until motion starts, and the last moment of stasis) | |

fs max = μsFn (at the moment motion starts) | |

fk = μkFn (when the two objects are sliding past each other - wheels don’t slide unless they are skidding). |

## Work

2 Ways to transfer Energy (J) between a system and its surroundings: Work and Heat. Work causes a shift in the center of mass (i.e.: a displacement). Heat does not. This is the basis of Thermodynamics

∆ Internal Energysys = Heaton sys + Workon sys

W = Fd cos θ

So, the cases where no work is done by a single force is when that Force is zero, no displacement occurs, or the angle between the force and the displacement vector θ is 90 or 270.

### Work-Energy Theorem

The formula for Kinetic Energy of an object:

KE = ½ mv2 (J = kg∙m2/s2)

Looking at the units, we can see that from the formula for the Work done by a constant Force,

W = Fdcos θ

(Units of work) = (N= kg∙m2/s2)(m) = (J = kg∙m2/s2)

Thus, we can see that Work must have to do with energy. In fact, Work is the transfer of kinetic energy into an object. If a force pulls kinetic energy out of a target object, then the sign of the Work done by that force will be negative.

Wnet = ∆ KE

If a motion does not cause a change in the speed, and thus the kinetic energy, of an object, no Work was done. Similarly, if there is a change in the Kinetic Energy, the sum of the Works done by all the forces acting on the object is nonzero, which is why I like to write the Work-Energy Theorem this way:

∑ Wby all forces = ∆ KE

## Mechanical Energy

Emech = K + U = KE + PE (Kinetic Energy + Potential Energy)

When conservative forces act on an object, they store the Kinetic energy lost as Potential Energy in the field of that force, thus they conserve the total mechanical energy.

### Law of the conservation of mechanical energy

Emech = K + U, thus:

∆Emech = ∆K + ∆U

Therefore, if Energy is conserved, the ∆Emech of our system is zero, because Emech is constant.:

∆Emech = 0 = ∆K + ∆U, thus:

∆K = -∆U

Which means that the potential energy lost is gained as kinetic energy. Which makes sense, using the example of gravitational potential energy: As an object falls, gravity does work on it, accelerating it, and speeding it up.

A “Newton’s Cradle.” Each ball on the end exchanges the kinetic energy transferred to it for gravitational potential energy, losing the kinetic energy (i.e.: slowing down) as it rises in height, gaining potential energy. After slowing down completely, the ball then accelerates back downwards, exchanging the gravitational potential energy for kinetic. The ball then transmits that energy via an impulse through the center 3 balls to the one on the other end. |

### Clinical applications of Mechanical Energy

Kinetic Energy: The higher the speed at which a car goes, the more deadly a crash will be. During the crash, the human body decelerates down to a speed of zero. Since KE = ½ mv2, going at “just” 78 mph (55 mph × √2) will lead to the driver hitting the car’s dashboard with twice as much Kinetic Energy. As I saw on a billboard in India, “Speed Thrills, But Kills!”

Potential Energy: The higher the height from which a person falls, the more danger they are in. The greater their initial height, the greater the magnitude of their ∆U, or loss of potential energy. By the law of conservation of mechanical energy, ∆K = - ∆U. Falls of more than 20ft (6.1m) need Trauma Center evaluation and care. Depending on the strength (Elastancee of the individual’s body, even lesser h. The rule of thumb is: 3 body heights = very bad.

The theme here is that the Energy of an impact is used to assess the mechanism of injury. Just in case you thought basic Physics had nothing to do with medicine :)

## Impulse

I = ∆ p = Fav ∆ t

Use impulse when a collision occurs, and they want you to figure out the average force exerted between the objects (or, if you know the mass, the acceleration), or the time.

Watch this video of Golf Ball smashing into steel plate: http://www.wimp.com/golfball/

Notice that the ball is elastic, which means stiff, i.e.: despite the stress of the collision, it has the same shape before and after. The kinetic energy the ball had before is used to deform it during the collision. Because highly elastic objects (rubber bands and bouncy balls) can resume their initial shapes, ALL of the energy comes back to the ball as it speeds back up after the collision, this time in a different direction.

During a collsion, the Force exerted by the particles on each other varies with time:

Impulse (I) = Favt = ∆ p = the Area under the curve, which is calculable from observing the momentum of the particle before and after the collision. In reality, the usual force exerted by the particles is a much sharper spike, better represented by the scale used in graph on the right. (see the golf ball video)

In Principia Mathematica (in which Newton gave birth to Physics and Calculus), He actually wrote the Second Law thusly:

By substitution,

Which is the most commonly taught version. Notice that Newton’s initial formulation explains how impulse (∆p) and Force are related.