Friction

Friction is the force that opposes the relative motion or tendency toward such motion of two surfaces in contact. It is not a fundamental force, as it is made up of electromagnetic forces between atoms. When contacting surfaces move relative to each other, the friction between the two objects converts kinetic energy into thermal energy, or heat. Friction between solid objects and fluids (gases or liquids) is called drag. Friction in an electronic circuit is called resistance. Contrary to popular belief, sliding friction is not caused by surface roughness, but by chemical bonding between the surfaces - by the stickiness of the two surfaces.

Classical approximation

The classical approximation of the force of friction between two solid surfaces is known as Coulomb friction, named after Charles-Augustin de Coulomb. The equation is:

,

where-

μ is the coefficient of friction, which is an empirical property of the contacting materials,

N is the normal force exerted between the surfaces, and

F_f is either the force exerted by friction, or, in the case of equality, the maximum possible magnitude of this force.

For surfaces in relative motion, μ is the coefficient of kinetic friction (see below), the Coulomb friction is equal to F_f, and the frictional force on each surface is exerted in the direction opposite to its motion relative to the other surface.

For surfaces at rest relative to each other, μ is the coefficient of static friction (generally larger than its kinetic counterpart), the Coulomb friction may take any value from zero up to F_f, and the direction of the frictional force on a surface is opposite to the motion that surface would experience in the absence of friction. Thus, in the static case, the frictional force is exactly what it must be in order to prevent motion between the surfaces; it balances the net force tending to cause such motion. In this case, rather than providing an estimate of the actual frictional force, the Coulomb approximation provides a threshold value for this force, above which sliding would commence.

This approximation mathematically follows from the assumptions that surfaces are in atomically close contact only over a small fraction of their overall area, that this contact area is proportional to the normal force (until saturation, which takes place when all area is in atomic contact), and that frictional force is proportional to contact area. Such reasoning aside, however, the approximation is fundamentally an empirical construction. Rather than a physical law, it is a rule of thumb describing the approximate outcome of an extremely complicated physical interaction. The strength of the approximation is its simplicity and versatility--though in general the relationship between normal force and frictional force is not exactly linear (and so the frictional force is not entirely independent of the contact area of the surfaces), the Coulomb approximation is an adequate representation of friction for the analysis of many physical systems.

The coefficient of friction (also known as the frictional coefficient) is a dimensionless scalar value which describes the ratio of the force of friction between two bodies and the force pressing them together. The coefficient of friction depends on the materials used -- for example, ice on steel has a low coefficient of friction (the two materials slide past each other easily), while rubber on pavement has a high coefficient of friction (the materials do not slide past each other easily). Coefficients of friction range from near zero to greater than one - under good conditions, a tire on concrete may have a coefficient of friction of 1.7.

When the surfaces are adhesive, Coulomb friction becomes a very poor approximation (for example, Scotch tape resists sliding even when there is no normal force, or a negative normal force). In this case, the frictional force may depend strongly on the area of contact. Some drag racing tires are adhesive in this way.

The force of friction is always exerted in a direction that opposes movement (for kinetic friction) or potential movement (for static friction) between the two surfaces. For example, a curling stone sliding along the ice experiences a kinetic force slowing it down. For an example of potential movement, the drive wheels of an accelerating car experience a frictional force pointing forward; if they did not, the wheels would spin, and the rubber would slide backwards along the pavement. Note that it is not the direction of movement of the vehicle they oppose, it is the direction of (potential) sliding between tire and road.

The coefficient of friction is an empirical measurement -- it has to be measured experimentally, and cannot be found through calculations. Rougher surfaces tend to have higher values. Most dry materials in combination have friction coefficient values between 0.3 and 0.6. Values outside this range are rarer, but Teflon, for example, can have a coefficient as low as 0.04. A value of zero would mean no friction at all, an elusive property--even Magnetic levitation vehicles have drag. Rubber in contact with other surfaces can yield friction coefficients from 1.0 to 2.

Static friction

Static friction occurs when the two objects are not moving relative to each other (like a rock on a table). The coefficient of static friction is typically denoted as μ_s. The initial force to get an object moving is often dominated by static friction. The static friction is in most cases higher than the kinetic friction. Rolling friction occurs when one object "rolls" on another (like a car's wheels on the ground). This is classified under static friction because the patch of the tire in contact with the ground, at any point while the tire spins, is stationary relative to the ground. The coefficient of rolling friction is typically denoted as μ_r.

Limiting friction is the maximum value of static friction, or the force of friction that acts when a body is just on the verge of motion on a surface.

Kinetic friction

Kinetic (or dynamic) friction occurs when two objects are moving relative to each other and rub together (like a sled on the ground). The coefficient of kinetic friction is typically denoted as μ_k, and is usually less than the coefficient of static friction. From the mathematical point of view, however, the difference between static and kinetic friction is of minor importance: Let us have a coefficient of friction which depends on the sliding velocity and is such that its value at 0 (the static friction μ_s ) is the limit of the kinetic friction μ_k for the velocity tending to zero. Then a solution of the contact problem with such Coulomb friction solves also the problem with the original μ_k and any static friction greater than that limit.

Since friction is exerted in a direction that opposes movement, kinetic friction usually does negative work, typically slowing something down. There are exceptions however, if the surface itself is under acceleration. One can see this by placing a heavy box on a rug, then pulling on the rug quickly. In this case, the box slides backwards relative to the rug, but moves forward relative to the floor. Thus, the kinetic friction between the box and rug accelerates the box in the same direction that the box moves, doing positive work.

Examples of kinetic friction:

• Sliding friction is when two objects are rubbing against each other. Putting a book flat on a desk and moving it around is an example of sliding friction

• Fluid friction is the friction between a solid object as it moves through a liquid or a gas. The drag of air on an airplane or of water on a swimmer are two examples of fluid friction.

• Resistance is the atomic friction of an electronic circuit or device as an electric current flows through it.

Reducing friction

Devices

Devices such as tires, ball bearings or rollers can change sliding friction into a much smaller type of rolling friction. Many thermoplastic materials such as nylon, HDPE and PTFE are commonly used for low friction bearings. They are especially useful because the coefficient of friction falls with increasing imposed load.

Techniques

One technique used by railroad engineers is to back up the train to create slack in the linkages between cars. This allows the train engine to pull forward and only take on the static friction of one car at a time, instead of all cars at once, thus spreading the static frictional force out over time.

Lubricants

A common way to reduce friction is by using a lubricant, such as oil, water, or grease, which is placed between the two surfaces, often dramatically lessening the coefficient of friction. The science of friction and lubrication is called tribology. Lubricant technology is when lubricants are mixed with the application of science, especially to industrial or commercial objectives.

Superlubricity, a recently-discovered effect, has been observed in graphite: it is the substantial decrease of friction between two sliding objects, approaching zero levels (a very small amount of frictional energy would still be dissipated).

Lubricants to overcome friction need not always be thin, turbulent fluids or powdery solids such as graphite and talc; acoustic lubrication actually uses sound as a lubricant.

Energy of friction

According to the law of conservation of energy, no energy is destroyed due to friction, though it may be lost to the system of concern. Energy is transformed from other forms into heat. A sliding hockey puck comes to rest due to friction as its kinetic energy changes into heat. Since heat quickly dissipates, many early philosophers, including Aristotle, wrongly concluded that moving objects lose energy without a driving force.

When an object is pushed along a surface, the energy converted to heat is given by:

where

N is the normal force,

μ_k is the coefficient of kinetic friction,

x is the coordinate along which the object transverses.

Physical deformation is associated with friction. While this can be beneficial, as in polishing, it is often a problem, as the materials are worn away, and may no longer hold the specified tolerances.

The work done by friction can translate into deformation and heat that in the long run may affect the surface's specification and the coefficient of friction itself. Friction can in some cases cause solid materials to melt.

Dielectric

A dielectric, or electrical insulator, is a substance that is highly resistant to the flow of an electric current. Although a vacuum is also an excellent dielectric, the following discussion applies primarily to physical substances.

When a dielectric medium interacts with an applied electric field, charges are redistributed within its atoms or molecules. This redistribution alters the shape of an applied electrical field both inside the dielectric medium and in the region nearby.

When two electric charges move through a dielectric medium, the interaction energies and forces between them are reduced. When an electromagnetic wave travels through a dielectric, its speed decreases and its wavelength increases.

When an electric field is initially applied across a dielectric medium, a current flows. The total current flowing through a real dielectric is made up of two parts: a conduction and a displacement current. In good dielectrics, the conduction current will be extremely small. The displacement current can be considered the elastic response of the dielectric material to any change in the applied electric field. As the magnitude of the electric field is increased, a displacement current flows, and the additional displacement is stored as potential energy within the dielectric. When the electric field is decreased, the dielectric releases some of the stored energy as a displacement current. The electric displacement can be separated into a vacuum contribution and one arising from the dielectric by

where P is the polarization of the medium, E is the electric field, D is the electric flux density (or displacement), and χ its electric susceptibility. It follows that the relative permittivity and susceptibility of a dielectric are related, .

Dielectric constant

The dielectric constant is known to be measured by k, C=kC_o is a measure of the extent to which a substance concentrates the electrostatic lines of flux. It is the ratio of the amount of electrical energy stored in an insulator, when a static electric field is imposed across it, relative to vacuum (which has a dielectric constant of 1). The dielectric constant is also known as the static permittivity.

Complex permittivity in dielectrics

Apart from a vacuum, the response of normal dielectrics to external fields generally depends on the frequency of the field. This frequency dependence is because a material's polarization does not respond instantaneously to an applied field. The response must always be causal (arising after the applied field). For this reason permittivity is often treated as a complex function of the frequency of the applied field ω, . The definition of permittivity therefore becomes

where D₀ and E₀ are the amplitudes of the displacement and electrical fields, respectively, i is the imaginary unit. The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity or dielectric constant (also ):

At the high-frequency limit, the complex permittivity is commonly referred to as ε_∞. At the plasma frequency and above, dielectrics behave as ideal metals, with electron gas behavior. The static permittivity is a good approximation for altering fields of low frequencies, and as the frequency increases a measurable phase difference δ emerges between D and E. The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate fields strength (E₀), D and E remain proportional, and

Since the response of materials to alternating fields is characterized by a complex permittivity, it is natural to separate its real and imaginary parts, which is done by convention in the following way:.

In the equation above, is the imaginary part of the permittivity, which is related to the rate at which energy is absorbed by the medium (converted into thermal energy, etcetera). The real part of the permittivity, , is related to the refractive index of the medium.

The study of capacitance involves using two key equations, they are Q=VC and W=0.5xQV

Dielectrics in Parallel-Plate Capacitors

The electrons in the molecules shift toward the positively charged left plate. The molecules then create a leftward electric field that partially cancels the field created by the plates. (The air gap is shown for clarity; in real capacitor, the dielectric is usually in direct contact with the plates.)

Putting a dielectric material between the plates in a parallel plate capacitor causes an increase in the capacitance in proportion to k, the relative permittivity of the material:

where ε₀ is the permittivity of free space, A is the area covered by the capacitors, and d is the distance between the plates.

This happens because an electric field polarizes the bound charges of the dielectric, producing concentrations of charge on its surfaces that create an electric field opposed (antiparallel) to that of the capacitor. Thus, a given amount of charge produces a weaker electric field between the plates than it would without the dielectric, which reduces the electric potential. Considered in reverse, this argument means that, with a dielectric, a given electric potential causes the capacitor to accumulate a larger charge polarization.

Applications

The use of a dielectric in a capacitor presents several advantages. The simplest of these is that the conducting plates can be placed very close to one another without risk of contact. Also, if subjected to a very high electric field, any substance will ionize and become a conductor. Dielectrics are more resistant to ionization than dry air, so a capacitor containing a dielectric can be subjected to a higher operating voltage. Layers of dielectric are commonly incorporated in manufactured capacitors to provide higher capacitance in a smaller space than capacitors using only air or a vacuum between their plates, and the term dielectric refers to this application as well as the insulation used in power and RF cables.

Some practical dielectrics

Dielectric materials can be solids, liquids, or gases. In addition, a high vacuum can also be a useful, lossless dielectric even though its relative dielectric constant is only unity.

Solid dielectrics are perhaps the most commonly used dielectrics in electrical engineering, and many solids are very good insulators. Some examples include porcelain, glass, and most plastics. Air, nitrogen and sulfur hexafluoride are the three most commonly used gaseous dielectrics.

• Industrial coatings such as parylene provide a dielectric barrier between the substrate and its environment.
• Mineral oil is used extensively inside electrical transformers as a fluid dielectric and to assist in cooling. Dielectric fluids with higher dielectric constants, such as electrical grade castor oil, are often used in high voltage capacitors to help prevent corona discharge and increase capacitance.
• Because dielectrics resist the flow of electricity, the surface of a dielectric may retain stranded excess electrical charges. This may occur accidentally when the dielectric is rubbed (the triboelectric effect). This can be useful, as in a Van de Graaff generator or electrophorus, or it can be potentially destructive as in the case of electrostatic discharge.
• Specially processed dielectrics, called electrets, may retain excess internal charge or "frozen in" polarization. Electrets have a semipermanent external electric field, and are the electrostatic equivalent to magnets. Electrets have numerous practical applications in the home and industry.
• Some dielectrics can generate a potential difference when subjected to mechanical stress, or change physical shape if an external voltage is applied across the material. This property is called piezoelectricity. Piezoelectric materials are another class of very useful dielectrics.
• Some ionic crystals and polymer dielectrics exhibit a spontaneous dipole moment which can be reversed by an externaly applied electric field. This behavior is called the ferroelectric effect. These materials are analogous to the way ferromagnetic materials behave within an externally applied magnetic field. Ferroelectric materials often have very high dielectric constants, making them quite useful for capacitors.

Pseudo Force

A fictitious force, also called a pseudo force^[1], is an apparent force that acts on all masses in a non-inertial frame of reference, e.g., a rotating reference frame. The force F does not arise from any physical interaction, but rather from the acceleration a of the non-inertial reference frame itself. Due to Newton's second law F = ma, fictitious forces are always proportional to the mass m being acted upon.

Role as calculational tool

It is sometimes convenient to solve physical problems in a non-inertial reference frame. In such cases, it is necessary to introduce fictitious forces to account for the acceleration of the reference frame. For example, the surface of the Earth is a rotating reference frame. To solve classical mechanics problems exactly in an Earth-bound reference frame, two fictitious forces must be introduced, the Coriolis force and the centrifugal force (described below), of which the Coriolis force is dominant on Earth. Both of these fictitious forces are weak compared to most typical forces in everyday life, but they can be detected under careful conditions. For example, Léon Foucault was able to show the Coriolis force that results from the Earth's rotation using the Foucault pendulum. If the Earth were to rotate a thousandfold faster (making each day only ~86 seconds long), people could easily get the impression that such fictitious forces are pulling on them, as on a spinning carousel.

Detection of non-inertial reference frame

Philosophers^{[attribution needed]} sometimes conjecture that a person living inside a closed box that is rotating (or otherwise accelerating) could not detect their own acceleration (rotation is a form of acceleration). That is not true, a person could not detect their velocity, but they could detect their acceleration (their change in velocity). Careful observers within the box can detect that they are in a non-inertial reference frame from the fictitious forces that arise from the acceleration of the box. They can even map out the magnitude and direction of the acceleration at every point within the box. The method by which this can be done is by detecting the movement of the box in a different rate/direction than their own for at least a short while.

This is similar to the lurch you would feel if you are driving and the driver suddenly speeds up: for a short while during and after the acceleration, the car is moving faster than you. Once your body catches up through the force of the seat pushing forward against your thigh or back, you stop feeling the lurch or shift (which is, in actuality, the force acting upon you). Another example of the detection of a non-inertial reference frame is the way a Foucault pendulum in a science museum will precess in exactly the same manner, regardless of whether the museum has walls or not.

In the box example, a person standing in the center of a very large box (for demonstration purposes, let's assume the box is large enough that they have some space around them and they're not touching a wall) has a velocity V, which is the same velocity of the box. If the box accelerated backwards rapidly, their feet would accelerate backwards with the box, faster than their body would accelerate, and they would trip forward. Although it might seem to those untrained in physics that a force had pushed him forward, that is an example of a fictitious force that relies on the movement of the box as a frame of reference (a frame of reference which is non-inertial). Someone with physics knowledge would probably be able to tell that the entire box had accelerated backwards, and their feet with it. A rule of thumb is that if you get lurched or turned in one direction, much like the man's feet were lurched backwards, it means the frame of reference you are in is accelerating in that direction.

Observers living inside a closed box that is moving uniformly (i.e., without acceleration) cannot detect their own motion. That is the essential physics of Newton's first two laws of motion.

Acceleration in a straight line

When a car accelerates hard, the common human response is to feel "pushed back into the seat." In an inertial frame of reference attached to the road, there is no physical force moving the rider backward. However, in the rider's non-inertial reference frame attached to the accelerating car, there is a backward fictitious force. We mention two possible ways of analyzing the problem:

1. From the viewpoint of an inertial reference frame with constant velocity matching the initial motion of the car, the car is accelerating. In order for the passenger to stay inside the car, a force must be exerted on him. This force is exerted by the seat, which has started to move forward with the car and compressed against the passenger until it transmits the full force to keep the passenger inside. Thus the passenger is accelerating in this frame, due to the unbalanced force of the seat.
2. From the point of view of the interior of the car, an accelerating reference frame, there is a fictitious force pushing the passenger backwards, with magnitude equal to the mass of the passenger times the acceleration of the car. This force pushes the passenger back into the seat, until the seat compresses and provides an equal and opposite force. Thereafter, the passenger is stationary in this frame, because the fictitious force and the (real) force of the seat are balanced.

This serves as an illustration of the manner in which fictitious forces arise from switching to a non-inertial reference frame. Calculations of physical quantities made in any frame give the same answers, but in some cases calculations are easier to make in a non-inertial frame. (In this simple example, the calculations are equally easy in either of the two frames described.)

Circular motion

A similar effect occurs in circular motion, circular for the standpoint of an inertial frame of reference attached to the road, with the fictitious force called the centrifugal force, fictitious when seen from a non-inertial frame of reference. If a car is moving at constant speed around a circular section of road, the occupants will feel pushed outside, away from the center of the turn. Again the situation can be viewed from inertial or non-inertial frames:

1. From the viewpoint of an inertial reference frame stationary with respect to the road, the car is accelerating toward the center of the circle. This is called centripetal acceleration and requires a centripetal force to maintain the motion. This force is maintained by the friction of the wheels on the road. The car is accelerating, due to the unbalanced force, which causes it to move in a circle.
2. From the viewpoint of a rotating frame, moving with the car, there is a fictitious centrifugal force that tends to push the car toward the outside of the road (and the occupants toward the outside of the car). The centrifugal force is balanced by the acceleration of the tires inward, making the car stationary in this non-inertial frame.

To consider another example, taking as our reference frame the surface of the rotating earth, centrifugal force reduces the apparent force of gravity by about one part in a thousand, depending on latitude. This is zero at the poles, maximum at the equator.

Another fictitious force that arises in the case of circular motion is the Coriolis force, which is ordinarily visible only in very large-scale motion like the projectile motion of long-range guns or the circulation of the earth's atmosphere. Neglecting air resistance, an object dropped from a 50 m high tower at the equator will fall 7.7 mm eastward of the spot below where it was dropped because of the Coriolis force.^[2]

In the case of distant objects and a rotating reference frame, what must be taken into account is the resultant force of centrifugal and coriolis force. Consider a distant star observed from a rotating spacecraft. In the reference frame co-rotating with the spacecraft the distant star appears to move along a circular trajectory around the spacecraft. The apparent motion of the star is an apparent centripetal acceleration. Just like in the example above of the car in circular motion, the centrifugal force has the same magnitude as the fictitious centripetal force, but is directed in the opposite, centrifugal direction. In this case the coriolis force is twice the magnitude of the centrifugal force, and it points in centripetal direction. The vector sum of the centrifugal force and the coriolis force is the total fictitious force, which in this case points in centripetal direction.

Fictitious forces and work

Fictitious forces can be considered to do work, provided that they move an object on a trajectory that changes its energy from potential to kinetic. For example, consider a person in a rotating chair holding a weight in his outstretched arm. If he pulls his arm inward, from the perspective of his rotating reference frame he has done work against centrifugal force. If he now lets go of the weight, from his perspective it spontaneously flies outward, because centrifugal force has done work on the object, converting its potential energy into kinetic. From an inertial viewpoint, of course, the object flies away from him because it is suddenly allowed to move in a straight line. This illustrates that the work done, like the total potential and kinetic energy of an object, can be different in a non-inertial frame than an inertial one.

Gravity as a fictitious force

All fictitious forces are proportional to the mass of the object upon which they act, which is also true for gravity. This led Albert Einstein to wonder whether gravity was a fictitious force as well. He noted that a freefalling observer in a closed box would not be able to detect the force of gravity; hence, free falling reference frames are equivalent to an inertial reference frame (the equivalence principle). Following up on this insight, Einstein was able to show (after ~9 years of work) that gravity is indeed a fictitious force; the apparent acceleration is actually inertial motion in curved spacetime. This is the essential physics of Einstein's theory of general relativity.

Mathematical derivation of fictitious forces

General derivation

Consider a particle with mass m and position vector x_a(t) in a particular inertial frame A. Consider a non-inertial frame B whose position relative to the inertial one is given by X(t). Since B is non-inertial, we must have that d²X/dt² (the acceleration of frame B with respect to frame A) is non-zero. Let the position of the particle in frame B be x_b(t). Then we have

Taking two time derivatives, this gives

Now consider the forces in the problem. By Newton's Second Law, F = ma. The true force is of course the one in frame A (the inertial one), so

However, suppose we are working to solve a problem in frame B. It may be useful to consider the apparent force in this frame, which is given by

Now we define

giving finally:

Thus we can solve problems in frame B by assuming that Newton's Second Law holds (with respect to quantities in that frame) and treating F_fictitious as an additional force.^[3]

Rotating coordinate systems

A common situation in which noninertial reference frames are useful is when the reference frame is rotating. Since such rotational motion is non-inertial, due to the acceleration present in any rotational motion, a fictitious force can always be invoked by using a rotational frame of reference. Despite this complication, the use of fictitious forces often simplifies the calculations involved.

The relationship between acceleration in an inertial frame, and that in a coordinate frame rotating with angular velocity can be expressed as

where we have used the relationship for the time derivative of a vector in rotating coordinates

, for any vector

Since , the acceleration becomes

or, equivalently,

The acceleration in the rotating frame equals

Since the force in the rotating frame is and, by definition, , the fictitious force equals

Here, the first term is the Coriolis force, the second term is the centrifugal force, and the third term is the Euler force.^[4] When the rate of rotation doesn't change, as is typically the case for a planet, the Euler force is zero.

Newton's Laws of Motion

Newton's First Law

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 tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction ( provided there is no unbalanced force acting along that direction) .

There are two parts to this statement – one which predicts the behavior of stationary objects and the other which predicts the behaviour of moving objects. These 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" (provided there is no 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."

There is a Pass the Water exercise that demonstrates this principle. If students participate in a relay race carrying a plastic container of water around a race track, the water will have a tendency to spill from the container at specific locations on the track. In general the water will spill when:

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

The behavior of the water during the relay race can be explained by Newton's first law of motion. The water spills whenever the state of motion of the container changes. The water resists this change in its own state of motion and tends to "keep on doing what it is doing." If the container is moved from rest to a high speed at the starting line; the water remains at rest and spills onto the table. When the container stops near the finish line; the water keeps moving and spills over the container's leading edge. If the container is forced to move in a different direction to make it around a curve; the water keeps moving in the original direction and spills over its edge.

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 tends to "keep 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 (which is 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 to move forward with the same speed and in the same direction, ultimately hitting the wind shield or the dashboard. Coffee in motion tends to stay 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 forward along the seat. A person in motion tends to stay in motion with the same speed and in the same direction ... unless if unbalanced force acts. 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 which 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 – try to provide explanations for each application.

• blood rushes from your head to your feet when riding on a descending elevator which suddenly stops.
• 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 the brick with a hammer. (CAUTION: Do not attempt this at home!)
• to dislodge ketchup from the bottom of a ketchup bottle, the bottle is often turned upside down, thrust downward at a high speed 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, a rock or another object which abruptly halts the motion of the skateboard.

Can you think of a few more examples which further illustrate applications of Newton's first law?

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/s². 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 net force increases, so will the object's acceleration. However, as the mass of the object increases, its acceleration will decrease.

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.

In terms of an equation, the net force is equal to the product of the object's mass and its acceleration.

F_net = m * a

Throughout this lesson, 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 is 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 "any 'ole force" in the above equation; it is the net force, not any of the individual forces, which is related to acceleration. As discussed earlier, 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.

The above equation also indicates that a unit of force is equal to a unit of mass multiplied by 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:

Thus, the definition of the standard metric unit of force is given 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/s².

The net force equation (F_net = m * a) can also be used as a "recipe" for algebraic problem-solving. The table below can be filled by substituting the known variables into the equation, then solving for the unknown quantity. Try it by yourself and then use the pop-up menus to view the answers.

	Net Force (N)	Mass (kg)	Acceleration (m/s2)
1.	10	2	Depress mouse. a = 5 m/s/s a = F/m a = 10 N/2 kg a = 5 m/s/s
2.	20	2	Depress mouse. a = 10 m/s/s a = F/m a = 20 N/2 kg a = 10 m/s/s
3.	20	4	Depress mouse. a = 5 m/s/s a = F/m a = 20 N/4 kg a = 5 m/s/s
4.	Depress mouse. F = 10 N F = m a F = 2 kg x 5 m/s/s F = 10 N	2	5
5.	10	Depress mouse. m = 1 kg m = F/a m = 10 N/10 m/s/s m = 1 kg	10

The numerical information in the table above demonstrates the important qualitative relationships between force, mass, and acceleration.

Comparing the values in rows 1 and 2, you see that doubling 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 "halving" the net force results in a "halving" of the acceleration (if mass is held constant). Acceleration is directly proportional to net force.

Observe from rows 2 and 3 that doubling the mass results in a "halving" of the acceleration (if force is held constant). And similarly, rows 4 and 5 show that "halving" the mass results in a doubling of the acceleration (if force is held constant). Acceleration is inversely proportional to mass.

Analysis of the data in the table illustrates that an equation such as F_net = m*a can used as a guide when thinking about how a variation in one quantity might affect another quantity. Whatever change is made of the net force, the same change will occur in the acceleration. Double, triple or quadruple the net force, and the acceleration will do the same. On the other hand, whatever change is made of the mass, the opposite or inverse change will occur in the acceleration. Double, triple or quadruple the mass, and the acceleration will be one-half, one-third or one-fourth of its original value.

As stated above, the direction of the net force is the same as the direction of the acceleration. Thus, if the direction of the acceleration is known, then so is the direction of the net force.

Consider the two ticker tape traces below which represent the acceleration of a car. From the traces, determine the direction of the net force upon the car. Then depress the mouse on the pop-up menus to view the answer. (Review acceleration.)

Depress mouse for answer. The net force is to the right since the acceleration is to the right. An object which moves to the right and speeds up has a rightward acceleration.
Depress mouse for answer. The net force is to the left since the acceleration is to the left. An object which moves to the right and slows down has a leftward acceleration.

In conclusion, Newton's second law explains the behavior of objects upon which unbalanced forces are acting. 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 of the object.

Check Your Understanding

1. What acceleration will result when a 12-N net force is applied to a 3-kg object? A 6-kg object?

Depress mouse to view answer. A 3-kg object experiences an accel'n of 4 m/s/s. A 6-kg object experiences an accel'n of 2 m/s/s.

2. A net force of 16 N causes a mass to accelerate at the rate of 5 m/s². Determine the mass.

Depress mouse to view answer. Use Fnet=m*a with Fnet=16 N and a=5 m/s/s. So (16 N) = (m)*(5 m/s/s) And m = 3.2 kg

3. An object is accelerating at 2 m/s². If the net force is tripled and the mass of the object is doubled, what is the new acceleration?

Depress mouse to view answer. Answer: 3 m/s/s. The original accel'n of 2 m/s/s must be multiplied by 3 (since a and F are directly proportional) and divided by 2 (since a and m are inversely proportional).

4. An object is accelerating at 2 m/s². If the net force is tripled and the mass of the object is halved, what is the new acceleration?

Depress mouse to view answer. Answer: 12 m/s/s. The original accel'n of 2 m/s/s must be multiplied by 3 (since a and F are directly proportional) and divided by 1/2 (since a and m are inversely proportional).

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Newton's Third Law

A force is a push or a pull upon an object which results from its interaction with another object. Forces result from interactions! Some forces result from contact interactions (normal, frictional, tensional, and applied forces are examples of contact forces) and other forces result from action-at-a-distance interactions (gravitational, electrical, and magnetic forces are examples of action-at-a-distance 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 force 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. In turn, the water reacts by 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 to 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 fishes to swim.

Consider the flying motion of birds. A bird flies by use of its wings. The wings of a bird push air downwards. In turn, the air reacts by 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 to 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 your automobile on your way to school. An automobile is equipped with wheels that spin backwards. As the wheels spin backwards, they push the road backwards. In turn, the road reacts by pushing the wheels forward. The size of the force on the road equals the size of the force on the wheels (or automobile); the direction of the force on the road (backwards) is opposite to 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 automobiles to move.

Check Your Understanding

1. While driving, Anna Litical observed a bug striking the windshield of her car. Obviously, a case of Newton's third law of motion. The bug hit the windshield and the windshield hit the bug. Which of the two forces is greater: the force on the bug or the force on the windshield?

Depress mouse to see answer. Trick Question! The forces are equal in size. For every action, there is an equal ... (equal!) reaction. The fact that the bug splatters only means that with its smaller mass, it is less able to withstand the larger acceleration resulting from the interaction.

2. Rockets are unable to accelerate in space because ...

1. there is no air in space for the rockets to push off of.
2. there is no gravity is in space.
3. there is no air resistance in space.
4. ... nonsense! Rockets do accelerate in space.

Depress mouse to see answer. The answer is D. It is a common misconception that rockets are unable to accelerate in space. Rockets do accelerate in space. Rockets are able to accelerate due to the fact that they burn fuel and push the exhaust gases in a direction opposite to the direction in which they wish to accelerate.

3. A gun recoils when it is fired. The recoil is the result of action-reaction force pairs. As the gases from the gunpowder explosion expand, the gun pushes the bullet forwards and the bullet pushes the gun backwards. The acceleration of the recoiling gun is ...

1. greater than the acceleration of the bullet.
2. smaller than the acceleration of the bullet.
3. the same size as the acceleration of the bullet.

Depress mouse to see answer. The answer is B. The force on the gun equals the force on the bullet. However, acceleration depends on both force and mass. The bullet has a greater acceleration due to the fact that it has a smaller mass. Remember: acceleration and mass are inversely proportional.

4. In the top picture, a physics student is pulling upon a rope which is attached to a wall. In the bottom picture, the physics student is pulling upon a rope which is held by the Strongman. In each case, the force scale reads 500 Newtons. The physics student is pulling

1. with more force when the rope is attached to the wall.
2. with more force when the rope is attached to the Strongman.
3. the same force in each case.

Depress mouse to see answer. The answer is C. The student is pulling with 500 N of force in each case. The rope transmits the force from the physics student to the wall (or to the Strongman) and vice versa. Since the force of the student pulling on the wall and the wall pulling on the student are action-reaction force pairs, they must have equal magnitudes. Inanimate objects such as walls can also push and pull.

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