chapter6

Chapter 6 Work and Energy

﻿Work & Energy Work (total) = Change in Kinetic Energy: This is the Work-Energy Theorem Work

A force does **work** if its point of application moves through a distance and there is a component of the force in the direction of the velocity of the force's point of application. If you push a cart to the right and it moves in that direction, you have done work on that car. The relationship between work and energy is that the **net work done on an object is equal to the change in the objects's kinetic energy (The Work-Energy Principle)**.

Here, this graph shows //force (parallel) vs. distance displaced.// The work done can be found simply by finding that area under the curve (or integrate the equation). This must be done to find the work done by a variable force like below!



Let's assume that the equation of the line is y = x^4/10 - 4x^3 + 5. To find the area under this curve, we must integrate this equation on the interval of 0 to 10. Thus, the work will look like this...



ANSWER IS NOOO!!!!!!!!! Dot Product

﻿When we want to find the dot product of two vectors (a and b), we simply mulitiply them together and by the cosine of the angle between the directions of the two values That means that if the vectors are both going in the exact opposite (cosine of 180 degrees), the dot product will be zero, and if in the same direction it will be 1 (cosine of 0 degrees).

Okay, so let's say that we have to use the dot product to find the total work done. How can we do this? We will take this problem for example.

Assume that the force here is 100N in the positive X direction, and this block has been pushed from the bottom to the top a distance of 10 meters. It has traveled 8 meters in the x-coordinates and 6 meters in the y-cooridinates. We will call the force "F" and this displacement "Z". To find the work done, we must start by taking the dot product of these two values. That means we must break down these values into their respective "i,j,k" values. F = 100i + 0j + 0k X = 8i + 6j + 0k The dot product equals (100N) * (8M) = 800 J

Two Types Of Energy

**Energy** is the capacity for doing work. When work is done by one system on another, energy is transferred between two systems. Kinetic energy is energy in motion, while potential energy is energy stored. There are many others, including electric, nuclear, thermal, and chemical energy. Kinetic ﻿When dealing with kinetic energy, we must specify whether it is translational (moving) or rotational (rolling). When the gun powder ignites, it transfers its chemical-potential energy into translational energy. The bullet is positioned in front of the explosion and is propelled forward with this translational kinetic energy.

Here are two problems; the first illustrates how one can find the work done on an object moving by a constant force, and the second shows how you can find work done by a varying force.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Here, a 50-kg box is moved across a level surface 40 m due to a constant force of 100 N. This force acts at an angle 37 degrees to the surface. (a) Find the work done by the force. (b) Find the work assuming that there is a friction force of 50N opposing the movement. (c) Find the net work done. (d) Would the total work done be effected by time?



<span style="color: #00ff00; font-family: 'Courier New',Courier,monospace; font-size: 150%;">Potential : <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Energy can be stored in many ways. One form of potential energy is gravity. If you place a ball on a table, it has potential energy relative to the ground beneath it. That is, if you pushed it off of the table, gravity would pull it down and give it velocity at the expense of its height (gravity potential). In this example, the string has been cocked back and therefore has potential energy due to a spring-like mechanism. If the woman were to release the arrow, the string would snap back towards its relaxed position and shoot the arrow in the process.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">If we assume that the spring starts unstretched, then we can disregard Uo (it equals 0). <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">We can also assume that the total work done on this system, if it oscillates, will always be ZERO. This is because the spring will pull it back to its initial position, compress, and re-displace it over and over again with no net displacement.

How can we find the work done by the spring on the interval of 0 to X1 ? It is simply the integral of the function with respect to x. Assume that the displacement is equal to 1 meter and the spring constant is 500 N/M.

<span style="color: #ff0000; font-family: Impact,Charcoal,sans-serif; font-size: 210%;">Forces

Conservative <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">A force is conservative if the total work it does on a particle is zero when the particle moves along any path and returns to its initial position. The work done by a conservative force on a particle is **INDEPENDENT OF THE PATH TAKEN** by the particle as it moves from one point another. Let's take gravity when skiing down a mountain, for example. When you ride the lift up the mountain, gravity is doing the work of -mgh (ignore the work done by the lift). Once you ski back down the mountain, gravity is the force pulling you down and therefore does the work of mgh. Assuming that you start and stop back in the same place, no net work has been done. Thus, gravity here is a conservative force.

Non-Conservative <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">On the other hand, a non-conservative force is one that adds nothing to the system and is **DIFFERENT BASED ON THE PATH** that a particle takes. The most relatable instance is when dealing with friction. If one were to move a crate across the room in the presence of friction, the amount of work varies based on the path taken. Asuming the friction is uniform, a straight path would require the least amount of work, whereas a zig-zag would require more. Both paths land you in the same position, but the zig-zag would require much more distance.

<span style="color: #00ff00; font-family: 'Courier New',Courier,monospace; font-size: 150%;">Common Examples
 * Conservative Forces || Nonconservative Forces ||
 * Gravitational || Friction ||
 * Elastic || Air Resistance ||
 * Elecrtic || Tension In A Chord ||
 * || Push Or Pull ||

<span style="color: #0000ff; font-family: Arial,Helvetica,sans-serif; font-size: 150%;">Power

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">The **power** supplied by a forces is the rate at which the force does work. It is equal to the derivitive of work, which is also equal to the dot product of the acting force and the instantaneous velocity. The unit for power is Watts

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">If you wanted to find how much power is required by something to achieve the work, such as a motor lifting up an object, you would execute this equation. Lets say the motor position directly overhead wants to lift a 500N object 5 meters straight up in distance, taking only 15 seconds.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">To calculate the required power, we would multiply the weight of the object, the distance displaced, and the cosine of the angle between the force and direction of movement, and finally divide that all by the time required. Our equation would look like

<span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif; font-size: 130%;">(500 * 5 * cos90 / 15) = 2500 / 15 = 166.66 Watts required

<span style="color: #ff0000; font-family: Impact,Charcoal,sans-serif; font-size: 200%;">Types of Equilibrium

Stable Equilibrium : <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Here, a small displacement in any direction results in a restoring force that accelerates the particle back toward its equilibrium position. In the electric field, an example would be a negative point charge placed in between two other negative charges. A displacement in either direction would result in the charge being pushed back towards the center point of equilibrium. Unstable Equilibrium : <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Here, a small displacement results in a force that accelerates the particle away from its equilibrium position. An example in an electric field would be a negative point charge sitting between two positive charges. A movement to any side would result it to be pushed back owards the middle distance between the two (equilibrium). Neutral Equilibrium :<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;"> Here, a small displacement results in zero force and the particle remains in equilibrium



Sources Used... AP Physics book for information Normal Physics book for information <span style="background-color: transparent; color: #000000; display: block; font-family: 'Times New Roman'; font-size: 16px; text-align: left; text-decoration: none;">http://www.golfranger.co.uk/images/work.gif <span style="background-color: transparent; color: #000000; display: block; font-family: 'Times New Roman'; font-size: 16px; text-align: left; text-decoration: none;">Table 6-1 AP Physics Book <span style="background-color: transparent; color: #810081; display: block; font-family: 'Times New Roman'; font-size: 16px; text-align: left;">__[]__ <span style="background-color: transparent; color: #810081; display: block; font-family: 'Times New Roman'; font-size: 16px; text-align: left;"><span style="background-color: transparent; color: #810081; display: block; font-family: 'Times New Roman'; font-size: 16px; text-align: left;">__[]__ []


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