Chapter 6
Other Forms of Energy
- Electric Energy - Nuclear Energy - Thermal Energy - Chemical Energy stored in food and fuels These forms of energy are considered to be part of kinetic and potential energy at the atomic or molecular level, rather than the mechanical. Work is done when energy is transferred from one object to another Accounting for all forms of energy, we find hat the total energy neither increases nor decreases. *energy is always conserved*
conservative force
A force, such as gravity, that performs work over a distance that is independent of the path taken. Rather it is dependent on the initial and final positions. it takes the same work (mgh) to lift an object of mass "m" vertically to a height "h" as to carry it up an incline of the same vertical height. Net work done by a conservative force on a particle moving around any closed loop is zero (round trip) the work done by a conservative force can be expressed as the difference in potential energy between initial and final positions.
Potential Energy Examples
A spring of a wind up toy. It acquired its potential energy because work was done on it by the person winding up the toy. As the spring unwinds, it exerts a force and does work to make the toy move. A stretched elastic band
Energy Conservation with Dissipative Forces
Because frictional forces reduce mechanical energy (but not the total energy), they are called dissipative forces If there is a non conservative force such as friction, and total energy is conserved, kinetic and potential energies become heat. The actual temperature rise of the materials involved can be calculated if heat is considered as a transfer of energy (thermal energy), then the total energy is conserved in any process.
Non conservative forces and potential energy
Because potential energy is energy associated with the position or configuration of objects, potential energy can only make sense if it can be stated uniquely for a given point. This cannot be done with nonconservative forces because the work done depends on the path taken. Hence, potential energy can be defined only for a conservative force. Thus, although potential energy is always associated with a force, not all forces have a potential energy. For example, there is no potential energy for friction.
Potential Energy of an Elastic Spring
Consider the simple coil spring. The spring has potential energy when compressed (or stretched), because when it is released,it can do work on a ball as shown. To hold a spring either stretched or compressed an amount "x" from its natural (unstretched) length requires the hand to exert an external force on the spring of magnitude which is directly proportional to x. *Fext* = kx
energy vs power
Energy - capacity to perform work Power - rate of change/production of energy, how quickly we perform work. A person may be able to walk a long distance before having to stop because so much energy has been expended. On the other hand, a person who runs very quickly may feel exhausted after only a few hundred meters. He or she is limited in this case by power, the rate at which his or her body can transform chemical energy into mechanical energy.
Law of Conservation of Energy
Energy cannot be created or destroyed The total energy is neither increased not decreased in any process. Energy can be transformed from one form to another, and transferred from one object to another, but the total amount remains constant.
nonconservative forces
Forces that its work depends on the path. Eg: friction, push, pull path dependent.
non conservative force example
Friction, Air resistance, Viscosity You push a crate across a floor from one point to another, the work you do depends on whether the path is straight or curved There is more work against friction if you push it in a curved path, rather than pushing it straight For the curved path the distance is greater, and unlike gravitational force, pushing force is in the direction of motion at each point.
Gravitational Potential energy "y"
Gravitational Potential Energy depends on the vertical height of the object above some reference point in some situations you may wonder from what point to measure height "y". The gravitational potential energy of a book held above a table depends on whether we measure y from the top of the table, the floor, or some other reference point. What is physically important in any situation is the change in potential energy, because that is what is related to the work done. We thus choose to measure y from any reference level that is convenient, but we must choose that start level and be consistent throughout. The change in potential energy between any two points does not depend on this choice.
Conservative Force Example
Gravitational, Electric, Elastic Elastic force of a spring or any elastic material in which F = -KX. Any object that starts at a given point and returns to the same point under the action of conservative force has no net work done on it because the potential energy is the same at the start and finish of the process. (round trip)
Mechanical Energy Conservation Example
If a rock starts from rest, all of the initial energy is potential energy. As the rock falls, the potential energy (mgy) decreases because the rock's height above the ground decreases. The rock's kinetic energy increases to compensate the loss of potential energy so that the sum of the two remain constant. An any point along the path the total mechanical energy is given by *E= KE + PE E = 1/2MV^2 + MGY*
Potential Energy as Stored Energy
In the above examples of potential energy—from a brick held at a height y, to a stretched or compressed spring—an object has the capacity or potential to do work even though it is not yet actually doing it. These examples show that energy can be stored, for later use, in the form of potential energy Note that there is a single universal formula for the translational kinetic energy of an object (1.2mv^2), but there is no single formula for potential energy. Instead, the mathematical form of the potential energy depends on the force involved.
Potential Energy of Spring
Potential Energy is stored in compressed and stretched strings the force increases as the spring is stretched or compressed further. the potential energy of the compressed/stretched string, measured from its equilibrium position, can be obtained by looking at the area under the F vs. X curve.
Work for an object that free falls
Potential energy converted to kinetic energy as the object falls. If we allow an object to start from rest at y2 and fall freely under the action of gravity, it acquires a velocity given by *v^2 = 2gh* after falling a height "h". it then has kinetic energy 1/2mv^2 = 1/2m(2gh) = mgh. And if it srikes a stake it can do work on the stake equal to mgh. Thus, to raise an object of a mass "m" to a height "h" requires an amount of work equal to mgh. And once at height "h", the object has the ability to do an amount of work equal to mgh. *we can say that the work done in lifting the object has been stored as gravitational potential energy*
Gravitational Potential to Kinetic Energy
Potential energy is converted to kinetic energy as an object falls Potential energy can be stores in a spring when it is compressed or stretched; when the spring is released, its potential energy can be converted to kinetic.
Gravitational Potential Energy
Potential energy that depends on the height of an object A heavy brick held high up above the ground has potential energy because of its position relative to the earth. The raised brick has the ability to do work, for if it is released, it will fall to the ground due to its gravitational force, and can do work on a stake, driving it into the ground.
Hooke's Law
The law stating that the stress of a solid is directly proportional to the strain applied to it. Fs = -kx the spring equation
Work-Energy Principle
The net work done on an object is equal to the change in the object's kinetic energy. The total net work Wnet is a sum of the work done by conservative forces WC and non conservative forces WNC WNET = WC + WNC The work WNC done by the nonconservative forces acting on an object is equal to the total change in kinetic and potential energies.
Conservation of Energy Examples
Two people ride down two different water slides that start at the same height. each rider's initial potential energy gets transformed into kinetic energy so the speed at the bottom is obtained from 1/2mv^2 = mgh. The masses cancel out so both people will reach the bottom of each their water slides at the same speed. since one water slide is steeper than the other, the rider on top of the steep slide will convert their potential energy into kinetic energy earlier. Therefore, this rider will travel faster than the other throughout the duration of the trip and reach the bottom first.
Principle of conservation of mechanical energy
The total mechanical energy of a system represent a conserved quantity The total mechanical energy E remains constant as long as no non conservative forces do work: KE + PE at some initial time 1 is equal to the KE + PE at any later time. *if only conservative forces do work, the total mechanical energy of a system neither increases no decreases in any process. It stays constant - or is conserved.
Work for a raised object
The two forces acting on the object are in opposite directions, so Fext = -Fg We can calculate the work done by each of these forces on the object by using our equation for work (see packet) Note that for height we can plug in (y2 - y1). y2 is the higher position, y1 is the lower position Even though the starting and ending positions are different, the displacements are equal, and therefore, so is the work done.
Change in Potential Energy
The work required of an external force to move the object *without acceleration* between two points the change in potential energy associated with a particular force is equal to the negative of the work done by that force when the object is moved from one point to a second.
Conservation of mechanical energy rules
This rule can only be applied to any object moving without friction under the action of gravity. A roller coaster car starting from rest at the top of a hill and coasting without friction to the bottom and up the hill on the other side if there is no friction the speed of the roller coaster will depend only on its height compared to the starting height. in a roller coaster question we cannot use the acceleration equations previously used in earlier chapters. The motion is not vertical, rather horizontal and a is not constant on the curved track.
work done by a constant force
W=Fdcos(theta) a force can be exerted onto an object, yet do no work on it. if you hold a heavy bag of groceries in your hand you exert a force on it, but there is no displacement of the bag since you are standing still. Additionally, when a force is perpendicular to displacement no work is done as well. For example, if you walk with the bag at a constant speed, no work is done. If you begin to run with the bag, the moment you change your speed from walking to running there is a horizontal acceleration which results in work being done. *work = force & displacement*
gravitational potential energy
We therefore define the gravitational potential energy of an object due to Earth's gravity, as *the product of the object's weight mg and its height "y" above some reference level (the ground)* the higher an object is above the ground, the more gravitational potential energy the change in potential energy when an object moves from a height y1 to a height y2 is equal to the work done by a net external force to move the object from position 1 to 2 without any acceleration Equivalently, we can define the change in gravitational potential energy in terms of the work done by gravity itself.
Raising an Object
When an object is raised by a hand at constant velocity, or no acceleration there are two forces acting upon it. The force of gravity acts downward and the force of the hand on the object counters the force of gravity, the is called the *extension force Fext*
Other acting forces
Work done by forces that oppose the direction of motion, will be negative. Such as the force of friction. The direction of the frictional force is opposite the displacement of the object
centripetal force
a force that acts on a body moving in a circular path and is directed toward the center around which the body is moving. *they do no work, since they are always perpendicular to the direction of motion* cos(180) = 0
dissipative forces
forces that transfer energy which is wasted
total mechanical energy
if there are no non conservative forces, the sum of the changes in the kinetic energy and in the potential energy is zero. kinetic and potential energy changes are equal but opposite in sign. This allows us to define total mechanical energy *E = KE + PE* * E2 = E1*
Solving problems with power
it is often convenient to write power in terms of net force F applied to an object and its speed v. P = W/t W = Fd Fd = force of distance traveled.
Potential Energy
stored energy that results from the position or shape of an object The energy associated with forces that depend on the position or configuration of an object (or objects) relative to the surroundings.
Conservation of energy elastic force (pole vault)
the initial kinetic energy of the running athlete is transformed into elastic potential energy of the bending pole. As the athlete leaves the ground, into gravitational potential energy. When the vaulter reaches the top and the pole has straightened out again, all of the energy is not gravitational potential energy. The pole doesn't supply energy, it stores is.
Power
the rate at which work is done The rate at which energy is transformed power is also needed for acceleration and for moving against the force of gravity
Force of Spring
the stretched of compressed spring itself exerts a force called in the opposite direction on the hand *Fs* = -kx - k= spring stiffness constant the force is sometimes called a restoring force because the spring exerts its force in the direction opposite the displacement, acting to return to its natural length.
Net work
work done by the net force, or vector sum of all the forces, acting on an object if you lift a rock up in the air the net work is equal to the work done by your hand, and the work done by gravity Work(hand) = Force*d*cos(theta) - mgycos(theta) Work (gravity) = Force*d*cos(theta) - mgycos(theta) *net work always equals zero*