Newton's Second Law

Newton's first law of motion predicts the behavior of objects for which all existing forces are balanced. The first law - sometimes referred to as the law of inertia - states that if the forces acting upon an object are balanced, then the acceleration of that object will be 0 m/s/s. Objects at equilibrium (the condition in which all forces balance) will not accelerate. According to Newton, an object will only accelerate if there is a net or unbalanced force acting upon it. The presence of an unbalanced force will accelerate an object - changing its speed, its direction, or both its speed and direction.
Newton's second law of motion pertains to the behavior of objects for which all existing forces are not balanced. The second law states that the acceleration of an object is dependent upon two variables - the net force acting upon the object and the mass of the object. The acceleration of an object depends directly upon the net force acting upon the object, and inversely upon the mass of the object. As the force acting upon an object is increased, the acceleration of the object is increased. As the mass of an object is increased, the acceleration of the object is decreased.

 

Newton's second law of motion can be formally stated as follows:

The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.

This verbal statement can be expressed in equation form as follows:

a = F_net / m

The above equation is often rearranged to a more familiar form as shown below. The net force is equated to the product of the mass times the acceleration.

F_net = m * a

In this entire discussion, the emphasis has been on the net force. The acceleration is directly proportional to the net force; the net force equals mass times acceleration; the acceleration in the same direction as the net force; an acceleration is produced by a net force. The NET FORCE. It is important to remember this distinction. Do not use the value of merely "any 'ole force" in the above equation. It is the net force that is related to acceleration. As discussed in an earlier lesson, the net force is the vector sum of all the forces. If all the individual forces acting upon an object are known, then the net force can be determined. If necessary, review this principle by returning to the practice questions in Lesson 2.

Consistent with the above equation, a unit of force is equal to a unit of mass times a unit of acceleration. By substituting standard metric units for force, mass, and acceleration into the above equation, the following unit equivalency can be written.
The definition of the standard metric unit of force is stated by the above equation. One Newton is defined as the amount of force required to give a 1-kg mass an acceleration of 1 m/s/s.
The F_net = m • a equation is often used in algebraic problem solving. The table below can be filled by substituting into the equation and solving for the unknown quantity. Try it yourself and then use the click on the buttons to view the answers.

	Net Force (N)	Mass (kg)	Acceleration (m/s/s)
1.	10	2
2.	20	2
3.	20	4
4.		2	5
5.	10		10

The numerical information in the table above demonstrates some important qualitative relationships between force, mass, and acceleration. Comparing the values in rows 1 and 2, it can be seen that a doubling of the net force results in a doubling of the acceleration (if mass is held constant). Similarly, comparing the values in rows 2 and 4 demonstrates that a halving of the net force results in a halving of the acceleration (if mass is held constant). Acceleration is directly proportional to net force.
Furthermore, the qualitative relationship between mass and acceleration can be seen by a comparison of the numerical values in the above table. Observe from rows 2 and 3 that a doubling of the mass results in a halving of the acceleration (if force is held constant). And similarly, rows 4 and 5 show that a halving of the mass results in a doubling of the acceleration (if force is held constant). Acceleration is inversely proportional to mass.
The analysis of the table data illustrates that an equation such as F_net = m*a can be a guide to thinking about how a variation in one quantity might effect another quantity. Whatever alteration is made of the net force, the same change will occur with the acceleration. Double, triple or quadruple the net force, and the acceleration will do the same. On the other hand, whatever alteration is made of the mass, the opposite or inverse change will occur with the acceleration. Double, triple or quadruple the mass, and the acceleration will be one-half, one-third or one-fourth its original value.

As stated above, the direction of the net force is in the same direction as the acceleration. Thus, if the direction of the acceleration is known, then the direction of the net force is also known. Consider the two oil drop diagrams below for an acceleration of a car. From the diagram, determine the direction of the net force that is acting upon the car. Then click the buttons to view the answers. (If necessary, review acceleration from the previous unit.)

 

In conclusion, Newton's second law provides the explanation for the behavior of objects upon which the forces do not balance. The law states that unbalanced forces cause objects to accelerate with an acceleration that is directly proportional to the net force and inversely proportional to the mass.

Rocket Science!

NASA rockets (and others) accelerate upward off the launch pad as they burn a tremendous amount of fuel. As the fuel is burned and exhausted to propel the rocket, the mass of the rocket changes. As such, the same propulsion force can result in increasing acceleration values over time. Use the Rocket Science widget below to explore this effect.

Newton's Third Law

A force is a push or a pull upon an object that results from its interaction with another object. Forces result from interactions! As discussed in Lesson 2, some forces result from contact interactions (normal, frictional, tensional, and applied forces are examples of contact forces) and other forces are the result of action-at-a-distance interactions (gravitational, electrical, and magnetic forces). According to Newton, whenever objects A and B interact with each other, they exert forces upon each other. When you sit in your chair, your body exerts a downward force on the chair and the chair exerts an upward force on your body. There are two forces resulting from this interaction - a force on the chair and a force on your body. These two forces are called action and reaction forces and are the subject of Newton's third law of motion. Formally stated, Newton's third law is:

For every action, there is an equal and opposite reaction.

The statement means that in every interaction, there is a pair of forces acting on the two interacting objects. The size of the forces on the first object equals the size of the force on the second object. The direction of the force on the first object is opposite to the direction of the force on the second object. Forces always come in pairs - equal and opposite action-reaction force pairs.
A variety of action-reaction force pairs are evident in nature. Consider the propulsion of a fish through the water. A fish uses its fins to push water backwards. But a push on the water will only serve to accelerate the water. Since forces result from mutual interactions, the water must also be pushing the fish forwards, propelling the fish through the water. The size of the force on the water equals the size of the force on the fish; the direction of the force on the water (backwards) is opposite the direction of the force on the fish (forwards). For every action, there is an equal (in size) and opposite (in direction) reaction force. Action-reaction force pairs make it possible for fish to swim.
Consider the flying motion of birds. A bird flies by use of its wings. The wings of a bird push air downwards. Since forces result from mutual interactions, the air must also be pushing the bird upwards. The size of the force on the air equals the size of the force on the bird; the direction of the force on the air (downwards) is opposite the direction of the force on the bird (upwards). For every action, there is an equal (in size) and opposite (in direction) reaction. Action-reaction force pairs make it possible for birds to fly.
Consider the motion of a car on the way to school. A car is equipped with wheels that spin. As the wheels spin, they grip the road and push the road backwards. Since forces result from mutual interactions, the road must also be pushing the wheels forward. The size of the force on the road equals the size of the force on the wheels (or car); the direction of the force on the road (backwards) is opposite the direction of the force on the wheels (forwards). For every action, there is an equal (in size) and opposite (in direction) reaction. Action-reaction force pairs make it possible for cars to move along a roadway surface.
 

 

Check Your Understanding

1. While driving down the road, a firefly strikes the windshield of a bus and makes a quite obvious mess in front of the face of the driver. This is a clear case of Newton's third law of motion. The firefly hit the bus and the bus hits the firefly. Which of the two forces is greater: the force on the firefly or the force on the bus?

 

 

2. For years, space travel was believed to be impossible because there was nothing that rockets could push off of in space in order to provide the propulsion necessary to accelerate. This inability of a rocket to provide propulsion is because ...

a. ... space is void of air so the rockets have nothing to push off of. b. ... gravity is absent in space.
c. ... space is void of air and so there is no air resistance in space.
d. ... nonsense! Rockets do accelerate in space and have been able to do so for a long time.

 

 

3. Many people are familiar with the fact that a rifle recoils when fired. This recoil is the result of action-reaction force pairs. A gunpowder explosion creates hot gases that expand outward allowing the rifle to push forward on the bullet. Consistent with Newton's third law of motion, the bullet pushes backwards upon the rifle. The acceleration of the recoiling rifle is ...

a. greater than the acceleration of the bullet. b. smaller than the acceleration of the bullet.
c. the same size as the acceleration of the bullet.

  4. In the top picture (below), Kent Budgett is pulling upon a rope that is attached to a wall. In the bottom picture, the Kent is pulling upon a rope that is attached to an elephant. In each case, the force scale reads 500 Newton. Kent is pulling ...

a. with more force when the rope is attached to the wall. b. with more force when the rope is attached to the elephant.
c. the same force in each case.

 

 

polarization

In an earlier section of Lesson 1, it was stated that an electrical attraction would be observed between a charged object and a neutral object. If a charged plastic tube is held near to neutral paper bits, the attraction between the paper and the plastic would be sufficient to raise the paper off the table. If a rubber balloon is charged by rubbing it with animal fur, the balloon can subsequently be stuck to the surface of a wooden cabinet or a whiteboard. Quite surprisingly, this interaction between a neutral object and any charged object can be explained using our usual rules of opposites attract and likes repel.
As discussed previously, an atom consists of positively charged protons and negatively charged electrons. The protons are in the nucleus of the atom, tightly bound and incapable of movement. The electrons are located in the vast regions of space surrounding the nucleus, known as the electron shells or the electron clouds. Relative to the protons of the nucleus, these electrons are loosely bound. In conducting objects, they are so loosely bound that they may be induced into moving from one portion of the object to another portion of the object. To get an electron in a conducting object to get up and go, all that must be done is to place a charged object nearby the conducting object.
To illustrate this induced movement of electrons, we will consider an aluminum pop can that is taped to a Styrofoam cup. The Styrofoam cup serves as both an insulating stand and a handle. A rubber balloon is charged negatively, perhaps by rubbing it against animal fur. If the negatively charged balloon is brought near the aluminum pop can, the electrons within the pop can will experience a repulsive force. The repulsion will be greatest for those electrons that are nearest the negatively charged balloon. Many of these electrons will be induced into moving away from the repulsive balloon. Being present within a conducting material, the electrons are free to move from atom to atom. As such, there is a mass migration of electrons from the balloon's side of the aluminum can towards the opposite side of the can. This electron movement leaves atoms on the balloon's side of the can with a shortage of electrons; they become positively charged. And the atoms on the side opposite of the can have an excess of electrons; they become negatively charged. The two sides of the aluminum pop can have opposite charges. Overall the can is electrically neutral; it's just that the positive and negative charge has been separated from each other. We say that the charge in the can has been polarized.
In general terms, polarization means to separate into opposites. In the political world, we often observe that a collection of people becomes polarized over some issue. For instance, we might say that the United States has become polarized over the issue of the death penalty. That is, the citizens of the United States have been separated into opposites - those who are for the death penalty and those who are against the death penalty. In the context of electricity, polarization is the process of separating opposite charges within an object. The positive charge becomes separated from the negative charge. By inducing the movement of electrons within an object, one side of the object is left with an excess of positive charge and the other side of the object is left with an excess of negative charge. Charge becomes separated into opposites.
The polarization process always involves the use of a charged object to induce electron movement or electron rearrangement. In the above diagram and accompanying discussion, electrons within a conducting object were induced into moving from the left side of the conducting can to the right side of the can. Being a conductor, electrons were capable of moving from atom to atom across the entire surface of the conductor. But what if the object being polarized is an insulator? Electrons are not free to move across the surface of an insulator. How can an insulator such as a wooden wall be polarized?

How Can an Insulator be Polarized?

Polarization can occur within insulators, but the process occurs in a different manner than it does within a conductor. In a conducting object, electrons are induced into movement across the surface of the conductor from one side of the object to the opposite side. In an insulator, electrons merely redistribute themselves within the atom or molecules nearest the outer surface of the object. To understand the electron redistribution process, it is important to take another brief excursion into the world of atoms, molecules and chemical bonds.
The electrons surrounding the nucleus of an atom are believed to be located in regions of space with specific shapes and sizes. The actual size and shape of these regions is determined by the high-powered mathematical equations common to Quantum Mechanics. Rather than being located a specific distance from the nucleus in a fixed orbit, the electrons are simply thought of as being located in regions often referred to as electron clouds. At any given moment, the electron is likely to be found at some location within the cloud. The electron clouds have varying density; the density of the cloud is considered to be greatest in the portion of the cloud where the electron has the greatest probability of being found at any given moment. And conversely, the electron cloud density is least in the regions where the electron is least likely to be found. In addition to having varying density, these electron clouds are also highly distortable. The presence of neighboring atoms with high electron affinity can distort the electron clouds around atoms. Rather than being located symmetrically about the positive nucleus, the cloud becomes asymmetrically shaped. As such, there is a polarization of the atom as the centers of positive and negative charge are no longer located in the same location. The atom is still a neutral atom; it has just become polarized.
The discussion becomes even more complex (and perhaps too complex for our purposes) when we consider molecules - combination of atoms bonded together. In molecules, atoms are bonded together as protons in one atom attract the electrons in the clouds of another atom. This electrostatic attraction results in a bond between the two atoms. Electrons are shared by the two atoms as they begin to overlap their electron clouds. If the atoms are of different types (for instance, one atom is Hydrogen and the other atom is Oxygen), then the electrons within the clouds of the two atoms are not equally shared by the atoms. The clouds become distorted, with the electrons having the greatest probability of being found closest to the more electron-greedy atom. The bond is said to be a polar bond. The distribution of electrons within the cloud is shifted more towards one atom than towards the other atom. This is the case for the two hydrogen-oxygen bonds in the water molecule. Electrons shared by these two atoms are drawn more towards the oxygen atom than towards the hydrogen atom. Subsequently, there is a separation of charge, with oxygen having a partially negative charge and hydrogen having a partially positive charge.
It is very common to observe this polarization within molecules. In molecules that have long chains of atoms bonded together, there are often several locations along the chain or near the ends of the chain that have polar bonds. This polarization leaves the molecule with areas that have a concentration of positive charges and other areas with a concentration of negative charges. This principle is utilized in the manufacture of certain commercial products that are used to reduce static cling. The centers of positive and negative charge within the product are drawn to excess charge residing on the clothes. There is a neutralization of the static charge buildup on the clothes, thus reducing their tendency to be attracted to each other. (Other products actually use a different principle. During manufacturing, a thin sheet is soaked in a solution containing positively charged ions. The sheet is tossed into the dryer with the clothes. Being saturated with positive charges, the sheet is capable of attracting excess electrons that are scuffed off of clothes during the drying cycle.)

How Does Polarization Explain the Balloon and the Wall Demonstration?

A complete discussion of the world of atoms, molecules and chemical bonds is beyond the scope of The Physics Classroom. Nonetheless, a model of the atom as a distortable cloud of negative electrons surrounding a positive nucleus becomes essential to understanding how an insulating material can be polarized. If a charged object is brought near an insulator, the charges on that object are capable of distorting the electron clouds of the insulator atoms. There is a polarization of the neutral atoms. As shown in the diagrams below, the neutral atoms of the insulator will orient themselves in such a manner as to place the more attractive charge nearest the charged object. Once polarized in this manner, opposites can now attract.

A common demonstration performed in class involved bringing a negatively charged balloon near a wooden door or wooden cabinet. The molecules of wood will reorient themselves in such a way as to place their positive charges towards the negatively charged balloon. The distortion of their electron clouds will result in an alignment of the wood molecules in a manner that makes the wooden cabinet attracted to the negatively charged balloon. In human terms, one might say that the wood does some quick grooming and then places its most attractive side towards the balloon and its most repulsive side away from the balloon. In the world of static electricity, closeness counts. The negative balloon is closer to the positive portion of the wood molecules and further from the more repulsive negative portion. The balloon and the wall attract with sufficient force to cause the balloon to stick to the wall. From a mechanics standpoint, we would say that the balloon and the wall are pressed together with a large force. The large normal force on the balloon results in a large static friction force. This friction force balances the downward force of gravity and the balloon remains at rest.
Another common physics (and chemistry) demonstration involves using a charged object to deflect a stream of water from its path. Most often, a comb is charged negatively by combing one's hair or a rubber balloon is charged in a similar manner. The negatively charged object is then brought near to a falling stream of water, causing the stream to be attracted to the comb or balloon and alter its direction of fall. The demonstration illustrates the polar nature of water molecules. The hydrogen atoms serve as the positive poles within a water molecule; oxygen serves as the negative pole. Molecules of a liquid are free to rotate and move about; the water molecules realign themselves in order to put their positive poles towards the negatively charged object. Once polarized, the stream and the balloon (or comb) are attracted. As the water molecules within the stream fall past the balloon, this realignment of individual molecules happens quickly and the entire stream is deflected from its original downward direction.

Examples of the attraction between charged objects and neutral objects are numerous and often demonstrated by physics teachers. Paper bits become polarized and are attracted to a charged piece of acetate. Small penguins cut from a sheet of paper are attracted to a charged plastic golf tube and demonstrate their happy feet. A long wooden 2x4 is placed on a pivot and becomes polarized and attracted to a charged golf tube. To the astonishment of students, the force of attraction on the wood is large enough to rotate it about the pivot point.


Polarization is Not Charging
Perhaps the biggest misconception that pertains to polarization is the belief that polarization involves the charging of an object. Polarization is not charging! When an object becomes polarized, there is simply a redistribution of the centers of positive and negative charges within the object. Either by the movement of electrons across the surface of the object (as is the case in conductors) or through the distortion of electron clouds (as is the case in insulators), the centers of positive and negative charges become separated from each other. The atoms at one location on the object possess more protons than electrons and the atoms at another location have more electrons than protons. While there are the same number of protons and electrons within the object, these protons and electrons are not distributed in the same proportion across the object's surface. Yet, there are still equal numbers of positive charges (protons) and negative charges (electrons) within the object. While there is a separation of charge, there is NOT an imbalance of charge. When neutral objects become polarized, they are still neutral objects.
The process of polarization is often used in many charging methods. In one section of Lesson 2, the charging by induction process will be discussed. This charging process depends upon a charged object to induce polarization within a neutral object. While charging by induction includes polarization as one of its steps, polarization is still NOT a charging process. Details about the induction charging method can be read about in Lesson 2 of this unit.

Check Your Understanding

Use your understanding of charge to answer the following questions. When finished, click the button to view the answers.
1. A rubber balloon possesses a positive charge. If brought near and touched to the door of a wooden cabinet, it sticks to the door. This does not occur with an uncharged balloon. These two observations can lead one to conclude that the wall is _____.

a. electrically neutral	b. negatively charged
c. a conductor	d. lacking electrons

 

2. Which of the diagrams below best represents the charge distribution on a metal sphere when a positively charged plastic tube is placed nearby?


3. The distribution of electric charge in a H₂O molecule is nonuniform. The more electronegative oxygen atom attracts electrons from the hydrogen atom. Thus, the oxygen atoms acquire a partial negative charge and the hydrogen atoms acquire a partial positive charge. The water molecule is "polarized." Which diagram(s) below correctly portray(s) a pair of H₂O molecules? Explain.


4. True or False:

When an object becomes polarized, it acquires a charge and becomes a charged object.

 

5. Charged rubber rods are placed near a neutral conducting sphere, causing a redistribution of charge on the spheres. Which of the diagrams below depict the proper distribution of charge on the spheres? List all that apply.


6. In the above situation, the conducting sphere is ____. List all that apply.

a. charged

b. uncharged (neutral)

c. polarized

 

Jump To Lesson 2: Methods of Charging

 

Newton's Law

Newton's First Law

In a previous chapter of study, the variety of ways by which motion can be described (words, graphs, diagrams, numbers, etc.) was discussed. In this unit (Newton's Laws of Motion), the ways in which motion can be explained will be discussed. Isaac Newton (a 17th century scientist) put forth a variety of laws that explain why objects move (or don't move) as they do. These three laws have become known as Newton's three laws of motion. The focus of Lesson 1 is Newton's first law of motion - sometimes referred to as the law of inertia.
Newton's first law of motion is often stated as

An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

 

There are two parts to this statement - one that predicts the behavior of stationary objects and the other that predicts the behavior of moving objects. The two parts are summarized in the following diagram.

The behavior of all objects can be described by saying that objects tend to "keep on doing what they're doing" (unless acted upon by an unbalanced force). If at rest, they will continue in this same state of rest. If in motion with an eastward velocity of 5 m/s, they will continue in this same state of motion (5 m/s, East). If in motion with a leftward velocity of 2 m/s, they will continue in this same state of motion (2 m/s, left). The state of motion of an object is maintained as long as the object is not acted upon by an unbalanced force. All objects resist changes in their state of motion - they tend to "keep on doing what they're doing."

Suppose that you filled a baking dish to the rim with water and walked around an oval track making an attempt to complete a lap in the least amount of time. The water would have a tendency to spill from the container during specific locations on the track. In general the water spilled when:

• the container was at rest and you attempted to move it
• the container was in motion and you attempted to stop it
• the container was moving in one direction and you attempted to change its direction.

The water spills whenever the state of motion of the container is changed. The water resisted this change in its own state of motion. The water tended to "keep on doing what it was doing." The container was moved from rest to a high speed at the starting line; the water remained at rest and spilled onto the table. The container was stopped near the finish line; the water kept moving and spilled over container's leading edge. The container was forced to move in a different direction to make it around a curve; the water kept moving in the same direction and spilled over its edge. The behavior of the water during the lap around the track can be explained by Newton's first law of motion.

Everyday Applications of Newton's First Law

There are many applications of Newton's first law of motion. Consider some of your experiences in an automobile. Have you ever observed the behavior of coffee in a coffee cup filled to the rim while starting a car from rest or while bringing a car to rest from a state of motion? Coffee "keeps on doing what it is doing." When you accelerate a car from rest, the road provides an unbalanced force on the spinning wheels to push the car forward; yet the coffee (that was at rest) wants to stay at rest. While the car accelerates forward, the coffee remains in the same position; subsequently, the car accelerates out from under the coffee and the coffee spills in your lap. On the other hand, when braking from a state of motion the coffee continues forward with the same speed and in the same direction, ultimately hitting the windshield or the dash. Coffee in motion stays in motion.
Have you ever experienced inertia (resisting changes in your state of motion) in an automobile while it is braking to a stop? The force of the road on the locked wheels provides the unbalanced force to change the car's state of motion, yet there is no unbalanced force to change your own state of motion. Thus, you continue in motion, sliding along the seat in forward motion. A person in motion stays in motion with the same speed and in the same direction ... unless acted upon by the unbalanced force of a seat belt. Yes! Seat belts are used to provide safety for passengers whose motion is governed by Newton's laws. The seat belt provides the unbalanced force that brings you from a state of motion to a state of rest. Perhaps you could speculate what would occur when no seat belt is used.


 
There are many more applications of Newton's first law of motion. Several applications are listed below. Perhaps you could think about the law of inertia and provide explanations for each application.

• Blood rushes from your head to your feet while quickly stopping when riding on a descending elevator.
• The head of a hammer can be tightened onto the wooden handle by banging the bottom of the handle against a hard surface.
• A brick is painlessly broken over the hand of a physics teacher by slamming it with a hammer. (CAUTION: do not attempt this at home!)
• To dislodge ketchup from the bottom of a ketchup bottle, it is often turned upside down and thrusted downward at high speeds and then abruptly halted.
• Headrests are placed in cars to prevent whiplash injuries during rear-end collisions.
• While riding a skateboard (or wagon or bicycle), you fly forward off the board when hitting a curb or rock or other object that abruptly halts the motion of the skateboard.

 

Try This At Home

Acquire a metal coat hanger for which you have permission to destroy. Pull the coat hanger apart. Using duct tape, attach two tennis balls to opposite ends of the coat hanger as shown in the diagram at the right. Bend the hanger so that there is a flat part that balances on the head of a person. The ends of the hanger with the tennis balls should hang low (below the balancing point). Place the hanger on your head and balance it. Then quickly spin in a circle. What do the tennis balls do?

application of optics

Optics and Optical phenomena find many examples in nature such as the formation of the rainbow, the phenomenon of mirage and twinkling of stars. There are many other applications of optics using lens systems, mirrors, lasers and diffraction gratings etc. Many beautiful experiments can be designed in Optics and put to various uses.

The field of ray optics is used to design and use the Microscope, the Telescope and cameras. There are various designs of the microscope which give different levels of magnification and resolution. The telescope, similary has different types of designs. Optics is also used in the design of precision components and systems. Precise instruments are important in machine design and testing.

Other applications include using spectrometers to analyze the spectrum of a source, and hence to deduce its characteristics. This method is called spectroscopy. Spectroscopy can be used to analyze the atomic structure of atoms, or the composition if a source.

The field of fiber optics is advanced and still a large amount of research is going on in fiber optics. This field is useful in communication systems. Fibers form the backbone of many communication systems.

Hologrpahy is yet another field of optics which holds promises in data storage.

In this chapter, we will study the motion in two dimensions further. First, we will analyze the path that any object takes when it is thrown in the air. We will see that all falling objects follows the same path. Then, we will analyze the motion that repeats itself, such as the motion of the propeller.

1. Projectile Motion 2. Circular Motion 3. Chapter 6 Quiz

		Section 1. Projectile Motion Objects launched are called projectiles. The flight of baseballs and basketballs are some examples of projectile motions. Let's analyze the projectile motion by breaking down the forces acting on the object. Suppose a ball was thrown horizontally at the velocity of 5 m/s. Let's think about forces acting on x-direction, or horizontal direction, only. We know that the ball was thrown at the velocity of 5 m/s. Because no other force acts on the ball in the air, we know the velocity will stay constant (because the net force is zero). Therefore, after 5 seconds, the ball is d = vt = 5 m/s * 5s = 25 m 25 m away. After 10 seconds, 5 m/s * 10 s = 50 m The ball is 50 m away.
		QUESTION: How far will the ball travel in 3 seconds in horizontal direction?

circular motion

In physics, circular motion is rotation along a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation (and thus constant speed), or non-uniform, that is, with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations describing circular motion of an object do not take size or geometry into account, rather, the motion of a point mass in a plane is assumed. In practice, the center of mass of a body can be considered to undergo circular motion.
Examples of circular motion include: an artificial satellite orbiting the Earth in geosynchronous orbit, a stone which is tied to a rope and is being swung in circles (cf. hammer throw), a racecar turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.
Circular motion is accelerated even if the angular rate of rotation is constant, because the object's velocity vector is constantly changing direction. Such change in direction of velocity involves acceleration of the moving object by a centripetal force, which pulls the moving object toward the center of the circular orbit. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion.

Formulas for uniform circular motion

Figure 1: Vector relationships for uniform circular motion; vector Ω representing the rotation is normal to the plane of the orbit.

For motion in a circle of radius r, the circumference of the circle is C = 2π r. If the period for one rotation is T, the angular rate of rotation, also known as angular velocity, ω is:

The speed of the object traveling the circle is:

The angle θ swept out in a time t is:

The acceleration due to change in the direction is:

The vector relationships are shown in Figure 1. The axis of rotation is shown as a vector Ω perpendicular to the plane of the orbit and with a magnitude ω = dθ / dt. The direction of Ω is chosen using the right-hand rule. With this convention for depicting rotation, the velocity is given by a vector cross product as

which is a vector perpendicular to both Ω and r ( t ), tangential to the orbit, and of magnitude ω r. Likewise, the acceleration is given by

which is a vector perpendicular to both Ω and v ( t ) of magnitude ω |v| = ω² r and directed exactly opposite to r ( t ).^[1]

[edit] Constant speed

In the simplest case the speed, mass and radius are constant.
Consider a body of one kilogram, moving in a circle of radius one metre, with an angular velocity of one radian per second.

• The speed is one metre per second.
• The inward acceleration is one metre per square second.
• It is subject to a centripetal force of one kilogram metre per square second, which is one newton.
• The momentum of the body is one kg·m·s⁻¹.
• The moment of inertia is one kg·m².
• The angular momentum is one kg·m²·s⁻¹.
• The kinetic energy is 1/2 joule.
• The circumference of the orbit is 2π (~ 6.283) metres.
• The period of the motion is 2π seconds per turn.
• The frequency is (2π)⁻¹ hertz.
• From the point of view of quantum mechanics, the system is in an excited state having quantum number ~ 9.48×10³⁵.

Then consider a body of mass m, moving in a circle of radius r, with an angular velocity of ω.

• The speed is v = r·ω.
• The centripetal (inward) acceleration is a = r·ω ² = r ⁻¹·v ².
• The centripetal force is F = m·a = r·m·ω ² = r⁻¹·m·v ².
• The momentum of the body is p = m·v = r·m·ω.
• The moment of inertia is I = r ²·m.
• The angular momentum is L = r·m·v = r ²·m·ω = I·ω.
• The kinetic energy is E = 2⁻¹·m·v ² = 2⁻¹·r ²·m·ω ² = (2·m)⁻¹·p ² = 2⁻¹·I·ω ² = (2·I)⁻¹·L ² .
• The circumference of the orbit is 2·π·r.
• The period of the motion is T = 2·π·ω ⁻¹.
• The frequency is f = T ⁻¹ . (Frequency is also often denoted by the Greek letter ν, which however is almost indistinguishable from the letter v used here for velocity).
• The quantum number is J = 2·π·L h⁻¹

  Description of circular motion using polar coordinates

  

  

  Figure 2: Polar coordinates for circular trajectory. On the left is a unit circle showing the changes and in the unit vectors and for a small increment dθ in angle θ.
  During circular motion the body moves on a curve that can be described in polar coordinate system as a fixed distance R from the center of the orbit taken as origin, oriented at an angle θ (t) from some reference direction. See Figure 2. The displacement vector is the radial vector from the origin to the particle location:
  

  

  where is the unit vector parallel to the radius vector at time t and pointing away from the origin. It is handy to introduce the unit vector orthogonal to as well, namely . It is customary to orient to point in the direction of travel along the orbit.
  The velocity is the time derivative of the displacement:
  

  

  Because the radius of the circle is constant, the radial component of the velocity is zero. The unit vector has a time-invariant magnitude of unity, so as time varies its tip always lies on a circle of unit radius, with an angle θ the same as the angle of . If the particle displacement rotates through an angle dθ in time dt, so does , describing an arc on the unit circle of magnitude dθ. See the unit circle at the left of Figure 2. Hence:
  

  

  where the direction of the change must be perpendicular to (or, in other words, along ) because any change d in the direction of would change the size of . The sign is positive, because an increase in dθ implies the object and have moved in the direction of . Hence the velocity becomes:
  

  

  
  The acceleration of the body can also be broken into radial and tangential components. The acceleration is the time derivative of the velocity:
  

  

  

  The time derivative of is found the same way as for . Again, is a unit vector and its tip traces a unit circle with an angle that is π/2 + θ. Hence, an increase in angle dθ by implies traces an arc of magnitude dθ, and as is orthogonal to , we have:
  

  

  where a negative sign is necessary to keep orthogonal to . (Otherwise, the angle between and would decrease with increase in dθ.) See the unit circle at the left of Figure 2. Consequently the acceleration is:
  

  

  

  The centripetal acceleration is the radial component, which is directed radially inward:
  

  

  while the tangential component changes the magnitude of the velocity:
  

  

  Using complex numbers

  Circular motion can be described using complex numbers. Let the x axis be the real axis and the y axis be the imaginary axis. The position of the body can then be given as z, a complex "vector":
  

  

  where i is the imaginary unit, and
  

  

  is the angle of the complex vector with the real axis and is a function of time t. Since the radius is constant:
  

  

  where a dot indicates time differentiation. With this notation the velocity becomes:
  

  

  and the acceleration becomes:
  

  

  

  

  The first term is opposite to the direction of the displacement vector and the second is perpendicular to it, just like the earlier results shown before.
  

  References

  1. ^ Knudsen, Jens M.; Hjorth, Poul G. (2000). Elements of Newtonian mechanics: including nonlinear dynamics (3 ed.). Springer. p. 96. ISBN 3-540-67652-X., Chapter 5 page 96