PHYSICS: Conservative and Non-conservative Forces

When a conservative force does work, you can get that energy changes forms, whether a conservative force does work or non-conservative does work. But when a conservative force does work, it’s easy to get the energy out of the system. And when a non-conservative force does work, it’s much tougher to get the energy out of the system.

To determine whether an object has conservative force or non-conservative forces:

1.       The work done by a conservative force when displacing an object from point “a” to point “b” is independent of the path taken.

·         Friction is a non-conservative force.

2.     The work done by a conservative force over a closed path is always zero.

Example, in the problem of free fall (neglecting air drag) we have already studied, an object is thrown upward at a certain initial speed. It reaches a maximum height where its velocity is zero and then starts to descend with negative velocity. Clearly the body has lost all its kinetic energy once it has reached the maximum height, because it has zero velocity. However, as the objects falls back, when it reaches the ground again its speed is identical to its initial speed. What happened is that on the way up the force of gravity does a negative work on the object, while on the way down it does a positive work, giving back to the object the same energy it had taken away. Forces like this are called conservative, because their work is never wasted, it can always be recovered. Forces that are not conservative, whose work cannot be recovered, are called non-conservative. The best examples are friction and air drag. Non-conservative forces tend to disperse their work into forms that cannot be recovered by simply reversing the motion, such as heat and sound.

It is useful to recast the work-energy theorem in a new way based on the distinction of conservative and non-conservative forces. The theorem was: W = ΔKE

Now we can rewrite this as:

WNC + WC = ΔKE

because the net work is the sum of the work of conservative and non-conservative forces.

Understanding Conservative Forces

·         The object regains initial motion (kinetic energy) on return to initial position in a closed path motion.

·         Conservative force transfers energy "to" and "from" an object during a closed path motion in equal measure.

·         Conservative force transfers energy between kinetic energy of the object in motion and the potential energy of the system interacting with the object.

·         Work done by conservative force is equal to work done by it on reversal of motion.

·         Total work done by conservative force in a closed path motion is zero.

Understanding Non-conservative Forces

·         The speed and kinetic energy of the object on return to initial position are lesser than initial values in a closed path motion.

·         Non - conservative force does not transfers energy "from" the system "to" the object in motion.

·         Non - conservative force transfers energy between kinetic energy of the object in motion and the system via energy forms other than potential energy.

·         Total work done by non-conservative force in a closed path motion is not zero.

Sources: www.youtube.com/

http://cass246.ucsd.edu/

http://cnx.org/content/m14104/latest/

PHYSICS: Kinetic Energy

Kinetic energy is the energy in motion. An object that has motion – whether it is vertical or horizontal motion – has kinetic energy. These are many forms of kinetic energy – vibrational (the energy due to vibrational motion), rotational (due to rotational energy), and translational (due to motion from one location to another). Focusing on translational, the following equation is used to represent the kinetic energy (KE) of an object:

Where, m = mass of object

v = speed of the object

This equation reveals that the kinetic energy of an object is already proportional to the square of its speed. That means, for a twofold increase in speed, the kinetic energy will increase by a factor of four. For a threefold increase in speed, the kinetic energy will increase by a factor of nine. And for a fourfold increase speed, the kinetic energy will increase by a factor of sixteen. The kinetic energy is dependent upon the square of the speed.

Kinetic energy is a scalar quantity; it does not have a direction. The kinetic energy of an object is completely described by magnitude alone. The standard metric unit of measurement for kinetic energy is the Joule – one Joule is equivalent to 1kg*(m/s) ^2.

Equations:

Linear motion:

Rotational motion:

where, cor means center of rotation, and com is the center of the mass.

Conservation of energy:

Linear motion (relativistic):

where the final approximate equality holds for v << c.

Extended Explanation:

When a (non-relativistic) particle of mass moves with velocity, the particle’s kinetic energy is given by

The relationship between the momentum p and velocity is p = mv equation (1) can also be written

For a collection of particles (labeled by index) the total kinetic energy is given by

where m_(i) is the mass of the ith particle and v_{(i)} is the magnitude of the ith particle’s velocity.

For the case of a continuous distribution of particles

For the case of the rigid body v(r) = ω x r for constant ω, and the above equation reduces to

where

In many cases (e.g., cubic, spherical) the system is symmetric enough that

in which case the above equation for kinetic energy reduces to

Sources: www.physicsclassroom.com/class/energy/

www.physics forums.com/

PHYSICS: Potential Energy

Potential energy exits whenever an object which has mass has a position within a force field. An object can store energy as the result of its position. Potential energy is the stored energy of position possessed by an object.

Gravitational Potential Energy

-         Is the energy stored in an object as the result of its vertical position or height. The gravitational potential energy of the massive ball of a demolition machine is dependent on two variables – the mass of the ball and the height to which it is raised. There is a direct relation between gravitational potential energy and the mass of an object. More massive objects have greater gravitational potential energy. There is also a direct relation between gravitational potential energy and the height of an object. The higher that an object is elevated, the greater the gravitational potential energy. These relationships are expressed by the following equation:

m = mass of the object

h = height of the object

g = gravitational field strength (9.8 N/kg)

Elastic Potential Energy

-         Is the energy stored in elastic materials as the result of their stretching or compressing. The amount of elastic potential energy stored in such a device is related to the amount of stretch of the device – the more stretch, the more stored energy.

Springs are a special instance of a device that can store elastic potential energy due to either compression or stretching. A force is required to compress a spring; the more compression there is, the more force that is required to compress it further. For certain springs, the amount of force is directly proportional to the amount of stretch or compression (x); the constant of proportionality is known as the spring constant (k).

Such spring are said to follow Hooke’s law. If a spring is not stretched or compressed, then there is no elastic potential energy stored in it. The spring is said to be at its equilibrium position. The equilibrium position is the position that the spring naturally assumes when there is no force applied to it. In terms of potential energy, the equilibrium position could be called the zero-potential energy position. There is a special equation for springs that relates the amount of elastic potential energy to the amount of stretch (or compression) and the spring constant. The equation is:

k = spring constant

x = amount of compression

(relative to equilibrium position)

Source: www.physicsclassroom.com/class/energy/

PHYSICS: Gravitation

Gravitation in everyday life is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped. Most people are familiar with gravity as the reason behind things staying on the earth’s surface, or “what goes up, must come down,” but gravity actually has a much vaster significance. Gravity is responsible for the formation of our earth and all other planets and for the movement of all heavenly bodies. It is gravity that makes our planet revolves around the sun and the moon revolve around the earth.

Newton’s first law states that the force of gravity between two masses is directly proportional to the product of the two masses and inversely proportional to the square of the distance between them, or mathematically: F = G(m₁m₂/d²), where G is a constant. Newton’s second law states that gravitational force is equal to the product of a body’s mass and its acceleration, or F = ma.

Formulas:

Universal Law of Gravitation

Where m₁ and m₂ are the masses of any two objects under consideration and r₁ and r₂ are their respective position vectors.

Equation for the gravitational constant

Sir Isaac Newton:

Weight and the Gravitational Force

Sources: www.wisegeek.com/

www.sparknotes.com/physics/gravitational/

www.csep10.phys.utk.edu/

PHYSICS: Free Fall

A free falling object is an object that is falling under the sole influence of gravity. Any object that is being acted upon only by the force of gravity is said to be in a state of free fall. There are two important motion characteristics that are true of free-falling objects:

• Free-falling objects do not encounter air resistance.
• It accelerates downwards at a rate of 9.8 m/s².

Because free-falling objects are accelerating downwards at a rate of 9.8 m/s², a dot diagram of its motion would depict acceleration. The dot diagram at the right depicts the acceleration of a free-falling object. The position of the object at regular time intervals – say, 0.1 second is shown. The fact that the distance that the object travels every interval of time is increasing is a sure sign that the ball is speeding up as it falls downward. An object travels downward and speeds up, and then its acceleration is downward.

Uniform Gravitational Field without Air Resistance

Where:
     v₀ – is the initial velocity (m/s)

     v(t) – vertical velocity with respect to time (m/s)

     y₀ – initial altitude (m)

     y(t) – altitude with respect to time (m)

     t – time elapsed

     g – acceleration due to gravity

Problem ~

An object in free fall is said to have reached terminal velocity. If the air resistance becomes strong enough to counter act all gravitational acceleration, causing the object to fall at a constant speed. The exact value of the terminal velocity varies according to the shape of the object, but can be estimated for many objects at 100m/s. When a 10kg object has reached terminal velocity, how much power does the air resistance exert on the object?

Solution:
To solve this problem, we will use the equation P = Fv cos θ instead of the usual power equation, as we are given the velocity of the object. We merely need to calculate the force exerted on the object by the air resistance, and the angle between the force and the velocity of the object. Since the object has reached a constant speed, the net force on it must be zero. Since there are only two forces acting on the object, gravity and air resistance, the air resistance must be equal in magnitude and opposite in direction as the force of gravity. Thus, F_a = - F₆ = mg 98N , pointing upwards. Thus, the force applied by air resistance is anti-parallel to the velocity of the object. Thus:

Sources: www.physicsclassroom.com/

www.sparknotes.com/physics/workenergypower/

PHYSICS: Projectile Motion

Projectile motion refers to the motion of an object projected into the air at an angle. A projectile is an object upon which the only force acting is gravity. Objects launched are called projectiles.

History:

Niccolo Tartaglia shows it was realized that projectiles actually follow curved path. Yet no one knew what that path was, until Galileo Galilee. First, he reasoned that a projectile is not only influenced by one motion, but two. The motion that acts vertically is the force of gravity that pulls an object towards the earth at 9.8 m/s². While having this motion, the projectile is also moving forward, horizontally at the same time. And this horizontal motion is uniform and constant according to Galileo’s principle of inertia. This two independent motions of him work together to create a precise mathematical curve – mathematical shape, a shape from Greeks called parabola.

Parabolic motion has the following properties.

1. They are under constant acceleration.
2. There, exists a non-zero component which is perpendicular to the acceleration so as to produce a parabolic path. If this condition is not satisfied, the motion is actually a linear motion. 

Basic Equations:

Horizontal velocity = u cos θ

Vertical velocity = u sin θ

The horizontal and vertical displacements of the objects are:

            (x) Horizontal displacement at time =  

            (y) Vertical displacement at time =

To find the maximum height:

So from dy/dx = 0, we get u sin θ – gt = 0

Sub (1) into the vertical displacement equation:

Equation of Trajectory:

From above, we know that:

            Horizontal motion (x) =

                                 

            Vertical motion (y) =  

                   

Eliminating t, we get

                          

This equation can be transformed to a quadratic equation in tan a

FORM OF EQUATIONS	USES
	1.        Find out the position of an object at a particular time. 2.      Find out the time taken for an object to travel to particular position.
	1.        Find out the y-position by given other condition. 2.      Find out the x-position by solving this quadratic equation.
	1.        Find out the angle of projection. 2.      Determine the minimum velocity of projection to clear an obstacle.

Source: www.library.thinkquest.org/

Pay It Forward.

Watching this film all over again, the more I understood the sense of the story. And there’s this story I remember that is more likely to this, the movie Milk in the year 2008. The difference was that the lead character is more mature at the age of 40, and is homosexual (it’s actually a true story about an American politician, Harvey Milk). But the exceptional thing to these movies, they have the same objective. They want to let us know that being a kid (or even fully grown) is not a thing about changing the world.

You don’t need to grow up first to change the world or trying to return your childhood. You don’t have to because it isn’t about being old or young. It’s always within oneself. By putting your perception into action, could make a big change. Change in you, your family, friends, relatives, until to the world.

I do really love the story, as well as the individuality of the lead character and its teacher. The story starts with a question, “What does the world mean to you?” and the class thinks. Then Mr. Simonet asked again, “What does the world expect of you?” Trevor answered, “Nothing.” That line really knocked me; because really, I don’t even know what does the world expect from me! I don’t even know how I can contribute a thing to this world.

The world for me is a family, my family, other’s family. Well, that’s the point, I mean, beginning of being you, yourself, or being/having [molding] one’s individuality. A loving parents, courteous relatives, adorable cousins… a much supported childhood. How lucky am I to have them? Always ready to help me in times of trouble.

Isn’t that great to visualize the world like my family? Though I know that it’s not always perfect, there’s at times (and won’t just disappear) that problems will get a way in. But there are a lot of reasons to hold on, a lot of reasons to stay, and reasons to change. Change not only inside of you, but inside of your family. We’re having problems to think of solutions, not of problems again. [From somebody]

And this one’s strange… Mr. Simonet asked for an assignment: “Think of an idea to change our world – and put it into ACTION!” Why? He knows that his class thinks of the world expects nothing from them as they were just seventh graders. Then why would he give an assignment about changing the world? Yes, and there goes, “Weird… crazy… hard… bummer…” from his class. But he insisted, “How about possible?”

Yea! How about possible? – Gosh, I cannot just imagine if I had wits just like the scriptwriter of this movie. Brilliant isn’t she? Her words are well-dedicated. Why the scriptwriter? Because I learned from somebody that films doesn’t look appealing to watch (just) because of the directors, but it (also) do look excellent because of the words, delightful words from scriptwriters.

Anyways, let’s go back to the story. Ü Yea, it is really possible to change our world. We still have a chance, if we still have a hope. But changing doesn’t stop to (just) hoping. You want to change a thing because you were holding on to something, or somebody believed that you can.

Changing always starts/refers on what somebody’s thinking and tries to place it into action. That’s what an eleven year old, Trevor McKinney did. Out of curiosity, he tried and thinks of something a child like him could do that special thing. And there’s a time that he wants to give up because he thinks that “Pay It Forward” didn’t work. But it did work.

He (Trevor) started with a man in a junk namely, Jerry. Oh, I really don’t know what he did to that man. What words did he tell to him or how did he help him. Jerry has a messed up life, and Trevor gave him a chance to live and fixed it. He gave him money to buy clothes and shoes to get a job.

Getting along with the story, Jerry thought that he can’t achieve what he has to pay forward until he saw a woman who wants to commit suicide. Then he changed his thinking. “Cause I owe somebody a favor…” that’s his reason why he help that woman. But no, he helped the woman not because he have to or owe somebody a favor, but he helped the woman because he knows that’s the right thing to do. And he believes that helping is a way his life would be saved.

A messed up life changed into something new. Someone visualized his new saved life and turned back home. And to thank that person who saved him from drowning, he followed what that person wants to, pay it forward. He pursued it not because it must be done. He pursues paying it forward because he also thinks that it is possible to change the world.

“What did you ever do to change the world?” – Trevor to Mr. Simonet. Oh yes, what did I/can I ever do to change the world? Quite ironic… but I must tell myself to think. Think of, as a teenager, an individual, what thing can I contribute to change our world? Before I die, could I have done something that I could be proud of?