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Definitions

 can be defined as transfer of energy. In physics we say that work is done on an Work  can object when you transfer energy to that object. If one object transfers (gives) energy to a second object, then the first object does work on the second object. Work is the application of a force over a distance. Lifting a weight from the ground and  putting it on a shelf is a good example of work. The force is equal to the weight of the object, and the distance is equal to the height of the shelf (W= Fxd). Work-Energy Principle --The change in the kinetic energy of an object is equal to the net work done on the object. Energy can be defined as the capacity for doing work. The simplest case of mechanical ork is when an object is standing still st ill and we force it to move. The energy of a moving object is called kinetic energy. For an object of mass m, moving with velocity of magnitude v, this energy can be calculated from the formula E= 1/2 mv^2. Types of Energy

There are two types of energy in many forms: Kinetic Energy = Energy of Motion Potential Energy = Stored Energy Forms of Energy

Solar Radiation -- Infrared Heat, Radio Waves, Gamma Rays, Microwaves, Ultraviolet Light Atomic/Nuclear Energy -energy released in nuclear reactions. When a neutron splits an atom's nucleus into smaller pieces it is called fission. When two nuclei are joined together under millions of degrees of heat it is called fusion Electrical Energy --The generation or use of electric power over a period of time expressed in kilowatt-hours (kWh), megawatt-hours (NM) or gigawatt-hours (GWh). Chemical Energy --Chemical energy is a form of potential energy related to the  breaking and forming of chemical bonds. It is stored in food, fuels and batteries, and is released as other forms of energy during chemical reactions.

Mechanical Energy -- Energy of the moving parts of a machine. Also refers to movements in humans Heat Energy -- a form of energy that is transferred by a difference in temperature What is Power

Power is the work done in a unit of time. In other words, power is a measure of how ho w quickly work can be done. The unit of power is the Watt = 1 Joule/ 1 second. One common unit of energy is the kilowatt-hour (kWh). If we are using one kW of  power, a kWh of energy will last one hour. Calculating Work, Energy and Power

WORK = W=Fd Because energy is the capacity to do work , we measure energy and work in the same units (N*m or joules). POWER (P) is the rate of energy generation (or absorption) over time:P = E/t Power's SI unit of measurement is the Watt, representing the generation or absorption of energy at the rate of 1 Joule/sec. Power's unit of measurement in the English system is the horsepower, which is equivalent to 735.7 Watts.

Mechanical Energy -- Energy of the moving parts of a machine. Also refers to movements in humans Heat Energy -- a form of energy that is transferred by a difference in temperature What is Power

Power is the work done in a unit of time. In other words, power is a measure of how ho w quickly work can be done. The unit of power is the Watt = 1 Joule/ 1 second. One common unit of energy is the kilowatt-hour (kWh). If we are using one kW of  power, a kWh of energy will last one hour. Calculating Work, Energy and Power

WORK = W=Fd Because energy is the capacity to do work , we measure energy and work in the same units (N*m or joules). POWER (P) is the rate of energy generation (or absorption) over time:P = E/t Power's SI unit of measurement is the Watt, representing the generation or absorption of energy at the rate of 1 Joule/sec. Power's unit of measurement in the English system is the horsepower, which is equivalent to 735.7 Watts.

Types of energy

Energy and how it can change forms. Kinetic, potential, and chemical energy. Introduction What does it mean to have energy? Well, think about how you feel when you wake up in the morning. If you have lots of energy, that probably means you feel awake, ready to go, and able to do what needs to be done during the day. If you have no energy (maybe

 because you didn’t get your eight hours of sleep), then you may not feel like getting out of bed, moving around, or doing the things you need to do. While this definition of energy is an everyday one, not a scientific one, it actually has a lot in common with the more formal definition of energy (and can give you a helpful way to remember it). Specifically, energy is defined as the ability to do work  –  which,  which, for  biology purposes, can be thought of as the ability to cause some kind of change. Energy

can take many different forms: for instance, we’re all familiar with light, heat, and electrical energy.

Here, we’ll look at some types of energy that are particularly important in biological systems, including kinetic energy (the energy of motion), potential energy (energy due to  position or structure), and chemical energy (the potential energy of chemical bonds). Energy is never lost, but it can be converted from one of these forms to another.

Kinetic energy

When an object is in motion, there is energy associated with that object. Why should that  be the case? Moving objects are capable of causing a change, or, put differently, differently, of doing work. For example, think of a wrecking ball. Even a slow-moving wrecking ball can do a lot of damage to another object, such as an empty house. However, a wrecking ball that is not moving does not do any work (does not knock in any buildings).

The energy associated with an object’s motion is called

kinetic energy. A speeding

 bullet, a walking person, and electromagnetic radiation like light all all have kinetic energy. Another example of kinetic energy is the energy associated with the constant, random  bouncing of atoms or molecules. This is also called thermal energy –  the  the greater the

thermal energy, the greater the kinetic energy of atomic motion, and vice versa. The average thermal energy of a group of molecules is what we call temperature, and when

thermal energy is being transferred between two objects, it’s known as heat. Potential energy

Let’s return to our wrecking ball example. The motionless wrecking ball doesn’t have any kinetic energy. But what would happen if it were lifted two stories up with a crane and suspended above a car? In this case, the wrecking ball isn't moving, but there is, in fact, still energy associated with it. The energy of the suspended wrecking ball reflects its  potential to do work (in this case, damage). If the wrecking ball were released, it would

do work by making a pancake of someone’s poor car. And if t he ball is heavier, the energy associated with it will be greater.

Image of water held behind a dam.  Image credit: OpenStax Biology, "Dam," by Pascal. This type of energy is known as potential energy, and it is the energy associated with an object because of its position or structure. For instance, the energy in the chemical bonds of a molecule is related to the structure of the molecule and the positions of its atoms relative to one another. Chemical energy, the energy stored in chemical bonds, is thus

considered a form of potential energy. Some everyday examples of potential energy include the energy of water held behind a dam, or of a person about to skydive out of an airplane.

Energy conversions

An object's energy can be converted from one form t o another. For instance, let’s consider our favorite example, the wrecking ball. As the wrecking ball hangs motionless several stories up, it has no kinetic energy, but a lot of potential energy. Once it is released, its kinetic energy begins to increase because it builds speed due to gravity, while its potential energy begins to decrease, because it is no longer as far from the ground. Just before it hits the ground, the ball has almost no potential energy and a lot of kinetic energy.

Diagram of octane, ball-and-stick model of octane, and image of a speeding racecar.  Image credit: OpenStax Biology. Lower panel, "Car," is a modification of work by  Russell Trow. The same kinds of conversions are possible with chemical energy, and we see lots of examples of this in our day-to-day lives. For instance, octane, a hydrocarbon found in gasoline, has chemical energy (potential energy) due to its molecular structure, which is shown above. This energy can be released in a car engine when the gasoline combusts,  producing high-temperature gases that move the engine’s pistons and, ultimately, propel the car forward (kinetic energy)^11start superscript, 1, end superscript. Part of the chemical energy is converted to the kinetic energy of the car, while part is converted to thermal energy as heat emitted from the engine.

Energy can change forms in a similar way in living organisms. For instance, energy stored in bonds of the small molecule ATP (potential energy) can power the movement of a motor protein and its cargo along a microtubule track, or the contraction of muscle cells to move a limb (kinetic energy). What is conservation of energy?

Learn what conservation of energy means, and how it can make solving problems easier. What is the principle of conservation of energy?

In physics, the term conservation refers to something which doesn't change. This means that the variable in an equation which represents a conserved quantity is constant over time. It has the same value both before and after an event. There are many conserved quantities in physics. They are often remarkably useful for making predictions in what would otherwise be very complicated situations. In mechanics, there are three fundamental quantities which are conserved. These are energy, momentum and angular momentum. If you have looked at examples in other articles — for example, the kinetic energy of charging elephants — then it may surprise you that energy is a conserved quantity. After all, energy often changes in collisions. It turns out that there are a couple of key qualifying statements we need to add: 

 Energy, as we'll be discussing it in this article, refers to the total energy of a system. As objects move around over time, the energy associated with them —  e.g., kinetic, gravitational potential, heat — might change forms, but if energy is conserved, then the total will remain the same.

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Conservation of energy applies only to isolated systems. A ball rolling across a rough floor will not obey the law of conservation of energy because it is not isolated from the floor. The floor is, in fact, doing work on the ball through friction. However, if we consider the ball and floor together, then conservation of energy will apply. We would normally call this combination the ball-floor system. In mechanics problems, we are likely to encounter systems containing kinetic (E_K   EK E, start subscript, K, end subscript), gravitational potential energy (U_gU 

 g U, start subscript, g, end subscript), elastic — spring —   potential energy (U_sUsU, start subscript, s, end subscript), and heat (thermal energy) (E_H EH E, start subscript, H, end subscript). Solving such problems often begins by establishing conservation of energy in a system between some initial time — subscript i — and at some later time — subscript f. E_\mathrm{Ki} + U_\mathrm{gi} + U_\mathrm{si} = E_\mathrm{Kf} + U_\mathrm{gf} + U_\mathrm{sf} + E_\mathrm{Hf} E Ki+U gi+U si= E Kf+U gf+U sf+ E Hf Which could be expanded out as: \frac{1}{2}mv_i^2 + mgh_i + \frac{1}{2}kx_i^2 = \frac{1}{2}mv_f^2 + mgh_f + \frac{1}{2}kx_f^2 + E_\mathrm{Hf}21mvi2+mghi+21kxi2=21mvf 2+mghf +21kxf 2+ E Hf What do we mean by  system here?

In physics, system is the suffix we give to a collection of objects that we choose to model with our equations. If we are to describe the motion of an object using conservation of energy, then the system should include the object of interest and all  other objects that it interacts with. In practice, we always have to choose to ignore some interactions. When defining a system, we are drawing a line around things we care about and things we don't. The things we don't include are usually collectively termed the environment . Ignoring some of the environment will inevitably make our calculations less accurate. There is no indignity in doing this however. In fact, being a good physicist is often as much about understanding the effects you need to describe as it is about knowing which effects can  be safely ignored. Consider the problem of a person making a bungee jump from a bridge. At a minimum, the system should include the jumper, bungee, and the Earth. A more accurate calculation might include the air, which does work on the jumper via drag, or air resistance. We could go further and include the bridge and its foundation, but since we know that the  bridge is much heavier than the jumper, we can safely ignore this. We wouldn't expect the force of a decelerating bungee jumper to have any significant effect on the bridge, especially if the bridge is designed to bear the load of heavy vehicles.

There is always some tenuous level of interaction between even distant objects, so we need to choose the boundary of our system intelligently. There is always some tenuous level of interaction between even distant objects, so we need to choose the boundary of our system intelligently. What is mechanical energy?

Mechanical energy, E_M EM E, start subscript, M, end subscript, is the sum of the  potential energy and kinetic energy in a system.  E M= E P+ E K You'll remember from your earlier work on conservative forces that it is only conservative forces like gravity and the spring force that have potential energy associated with them. Nonconservative forces like friction and drag do not. We can always get back the energy that we put into a system via a conservative force. Energy transferred by nonconservative forces however is difficult to recover. It often ends up as heat or some other form which is typically outside the system — in other words, lost to the environment. What this means in practice is that the special case of conservation of mechanical energy is often more useful for making calculations than conservation of energy in general. Conservation of mechanical energy only applies when all forces are conservative. Luckily, there are many situations where nonconservative forces are negligible, or at least a good approximation can still be made when neglecting them. How can conservation of energy describe how objects move?

When energy is conserved, we can set up equations which equate the sum of the different forms of energy in a system. We then may be able to solve the equations for velocity, distance, or some other parameter on which the energy depends. If we don't know enough of the variables to find a unique solution, then it may still be useful to plot related variables to see where solutions lie. Consider a golfer on the moon — gravitational acceleration 1.625 m/s^22start superscript, 2, end superscript — striking a golf ball. By the way, Astronaut Alan Shepard actually did

this. The ball leaves the club at an angle of 45^\circ degree to the lunar surface traveling ∘

at 20 m/s both horizontally and vertically — total velocity 28.28 m/s. How high would the  golf ball go? We begin by writing down the mechanical energy: E_\mathrm{M} = \frac{1}{2} m v^2 + mgh E M=21mv2+mgh Applying the principle of conservation of mechanical energy, we can solve for the height hhh — note that the mass cancels out. \frac{1}{2} m v_i^2 = mgh_f+\frac{1}{2} m v_f^221mvi2=mghf +21mvf 2start fraction, 1, divided by, 2, end fraction, m, v, start subscript, i, end subscript, start superscript, 2, end superscript, equals, m, g, h, start subscript, f, end subscript, plus, start fraction, 1, divided by, 2, end fraction, m, v, start subscript, f, end subscript, start superscript, 2, end superscript \begin{aligned} h &= \frac{\frac{1}{2}v_i^2-\frac{1}{2}v_f^2}{g} \\ &=\frac{\frac{1}{2}(28.28~\mathrm{m/s})^2\frac{1}{2}(20~\mathrm{m/s})^2}{1.625~\mathrm{m/s^2}} \\ &= 123~\mathrm{m}\end{aligned}h= g 21vi2−21vf 2=1.625 m/s221(28.28 m/s)2−21(20 m/s)2 =123 m [How did we know the final speed was 20 m/s?] As you can see, applying the principle of conservation of energy allows us to quickly solve problems like this which would be more difficult if done only with the kinematic equations. Exercise 1: Suppose the ball had an unexpected collision with a nearby american flag

hoisted to a height of 2 m. How fast would it be traveling at the time of collision? [Show solution.] \frac{1}{2} m v_i^2 = mgh_f+\frac{1}{2} m v_f^2start fraction, 1, divided by, 2, end fraction, m, v, start subscript, i, end subscript, start superscript, 2, end superscript, equals, m, g, h, start subscript, f, end subscript, plus, start fraction, 1, divided by, 2, end fraction, m, v, start subscript, f, end subscript, start superscript, 2, end superscript

vf =v2i – 2  g  h−−−−−−−−−√=28.16 m/s ⋅

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Exercise 2: The image below shows a plot of the kinetic, gravitational potential and

mechanical energy over time during the flight of a small model rocket. Points of interest such as maximum height, apogee, and the time of motor stop, burnout, are noted on the  plot. The rocket is subject to several conservative and nonconservative forces over the course of the flight. Is there a time during the flight when the rocket is subject to only conservative forces? Why?

Energy transfer during the flight of a small model rocket [1]. Energy transfer during the flight of a small model rocket [1]. Why can perpetual motion machines never work?

The perpetual motion machine is a concept for a machine which continues its motion forever, without any reduction in speed. An endless variety of weird and wonderful machines have been described over the years. They include pumps said to run themselves via their own head of falling water, wheels which are said to push themselves around by means of unbalanced masses, and many variations of self-repelling magnets. Though often interesting curiosities, such a machine has never been shown to be  perpetual, nor could it ever be. In fact, even if such a machine were to exist, it wouldn't  be very useful. It would have no ability to do work. Note that this differs from the

concept of the over-unity machine, which is said to output more than 100% of the energy  put into it, in clear violation of the principle of conservation of energy. From the most basic principles of mechanics, there is nothing that strictly makes the  perpetual motion machine impossible. If a system could be fully isolated from the environment and subject to only conservative forces, then energy would be conserved and it would run forever. The problem is that in reality, there is no way to completely isolate a system and energy is never completely conserved within the machine. It is possible today to make extremely low friction flywheels which rotate in a vacuum for storing energy. Yet, they still lose energy and eventually spin down when unloaded, some over a period of years [2]. The earth itself, rotating on its axis in space is perhaps an extreme example of such a machine. Yet, because of interactions with the moon, tidal friction, and other celestial bodies, it too is gradually slowing. In fact, every couple of years, scientists have to add a leap second  to our record of time to account for variation in the length of day.

Momentum 

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Momentum Momentum and Impulse Connection Real-World Applications The sports announcer says, "Going into the all-star break, the Chicago White Sox have the momentum." The headlines declare "Chicago Bulls Gaining Momentum." The coach pumps up his team at half-time, saying "You have the momentum; the critical need is that you use that momentum and bury them in this third quarter." Momentum is a commonly used term in sports. A team that has the momentum is on  the move and is going to take some effort to stop. A team that has a lot of momentum is really on the move and is going to be hard to stop. Momentum is a physics term; it refers to the quantity of motion that an object has. A sports team that is on the move has the momentum. If an object is in motion (on the move) then it has momentum.

Momentum can be defined as "mass in motion." All objects have mass; so if an object is moving, then it has momentum - it has its mass in motion. The amount of momentum that an object has is dependent upon two variables: how much stuff  is moving and how fast the stuff  is moving. Momentum depends upon the variables mass and velocity. In terms of an equation, the momentum of an object is equal to the mass of the object times the velocity of the object. Momentum = mass • velocit y In physics, the symbol for the quantity momentum is the lower case p. Thus, the above equation can be rewritten as p=m•v where m is the mass and v is the velocity. The equation illustrates that momentum is directly proportional to an object's mass and directly proportional to the object's velocity. The units for momentum would be mass units times velocity units. The standard metric

unit of momentum is the kg•m/s. While the kg•m/s is the standard metric unit of momentum, there are a variety of other units that are acceptable (though not

conventional) units of momentum. Examples include kg•mi/hr, kg•km/hr, and g•cm/s. In each of these examples, a mass unit is multiplied by a velocity unit to provide a momentum unit. This is consistent with the equation for momentum.

Momentum as a Vector Quantity Momentum is a vector quantity. As discussed in an earlier unit, a vector quantity is a quantity that is fully described by both magnitude and direction. To fully describe the momentum of a 5-kg bowling ball moving westward at 2 m/s, you must include information about both the magnitude and the direction of the bowling ball. It is not enough to say that the ball has 10 kg•m/s of momentum; the momentum of the  ball is not fully described until information about its direction is given. The direction of the momentum vector is the same as the direction of the velocity of the ball. In a previous unit, it was said that the direction of the velocity vector is the same as the direction that an object is moving. If the bowling ball is moving

westward, then its momentum can be fully described by saying that it is 10 kg•m/s,

westward. As a vector quantity, the momentum of an object is fully described by both magnitude and direction.

The Momentum Equation as a Guide to Thinking From the definition of momentum, it becomes obvious that an object has a large momentum if both its mass and its velocity are large. Both variables are of equal importance in determining the momentum of an object. Consider a Mack truck and a roller skate moving down the street at the same speed. The considerably greater mass of the Mack truck gives it a considerably greater momentum. Yet if the Mack truck were at rest, then the momentum of the least massive roller skate would be the greatest. The momentum of any object that is at rest is 0. Objects at rest do not have momentum - they do not have any "mass in motion." Both variables - mass and velocity - are important in comparing the momentum of two objects. The momentum equation can help us to think about how a change in one of the two variables might affect the momentum of an object. Consider a 0.5-kg physics cart loaded with one 0.5-kg brick and moving with a speed of 2.0 m/s. The total mass of loaded  cart is 1.0 kg and its momentum is 2.0 kg•m/s. If the cart was instead loaded with three 0.5 -kg  bricks, then the total mass of the loaded  cart would be 2.0 kg and its momentum would  be 4.0 kg•m/s. A doubling of the mass results in a doubling of the mome ntum. Similarly, if the 2.0-kg cart had a velocity of 8.0 m/s (instead of 2.0

m/s), then the cart would have a momentum of 16.0 kg•m/s (instead of 4.0 kg•m/s). A quadrupling  in velocity results in a quadrupling  of the momentum. These two examples illustrate how the equation p = m•v serves as a "guide to thinking" and  not merely a " plug-and-chug  recipe for algebraic  problem-solving." Momentum and Impulse Connection

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Momentum Momentum and Impulse Connection Real-World Applications As mentioned in the previous part of this lesson, momentum is a commonly used term in sports. When a sports announcer says that a team has the momentum they mean that the team is really on the move and is going to be hard to stop. The term momentum is a physics concept. Any object with momentum is going to be hard to stop. To stop such an object, it is necessary to apply a force against  its motion for a given  period of time. The more momentum that an object has, the harder that it is to stop. Thus, it would require a greater amount of force or a longer amount of time or both to bring such an object to a halt. As the force acts upon the object for a given amount of time, the object's velocity is changed; and hence, the object's momentum is changed. The concepts in the above paragraph should not seem like abstract information to you. You have observed this a number of times if you have watched the sport of football. In football, the defensive  players apply a force for a given amount of time to stop the momentum of the offensive player who has the ball. You have also experienced this a multitude of times while driving. As you  bring your car to a halt when approaching a stop sign or stoplight, the brakes serve to apply a force to the car for a given amount of time to change the car's momentum. An object with momentum can be stopped if a force is applied against it for a given amount of time. A force acting for a given amount of time will change an object's momentum. Put another way, an unbalanced force always accelerates an object - either speeding it up or slowing it down. If the force acts opposite the object's motion, it slows the object down. If a force acts in the same direction as the object's motion, then the force speeds the object up. Either way, a force will change the velocity of an object. And if the velocity of the object is changed, then the momentum of the object is changed.

Impulse These concepts are merely an outgrowth of  Newton's second law as discussed in an earlier unit. Newton's second law (F net = m • a) stated that the acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to the mass of the object. When combined with the definition of acceleration (a = change in velocity / time), the following equalities result.

F=m•a or

F = m • ∆v / t If both sides of the above equation are multiplied by the quantity t, a new equation results.

F • t = m • ∆v This equation represents one of two primary principles to be used in the analysis of collisions during this unit. To truly understand the equation, it is important to understand its meaning in words. In words, it could be said that the force times the time equals the

mass times the change in velocity. In physics, the quantity Force • time is known as impulse. And since the quantity m•v is the momentum, the quantity m•Δv must be the change in momentum. The equation really says that the Impulse = Change in momentum

One focus of this unit is to understand the physics of collisions. The physics of collisions are governed by the laws of momentum; and the first law that we discuss in this unit is expressed in the above equation. The equation is known as the impulse-momentum change equation. The law can be expressed this way: In a collision, an object experiences a force for a specific amount of time that results in a change in momentum. The result of the force acting for the given amount of time is that the object's mass either speeds up or slows down (or changes direction). The impulse experienced by the object equals the change in momentum of the object. In equation

form, F • t = m • Δ v. In a collision, objects experience an impulse; the impulse causes and is equal to the change in momentum. Consider a football halfback running down the football field and encountering a collision with a defensive back. The collision would change the halfback's speed and thus his momentum. If the motion was represented by a ticker tape diagram, it might appear as follows:

At approximately the tenth dot on the diagram, the collision occurs and lasts for a certain amount of time; in terms of dots, the collision lasts for a time equivalent to approximately nine dots. In the halfback-defensive back collision, the halfback experiences a force that lasts for a certain amount of time to change his momentum. Since

the collision causes the rightward-moving halfback to slow down, the force on the halfback must have been directed leftward. If the halfback experienced a force of 800 N for 0.9 seconds, then we could s ay that the impulse was 720 N•s. This impulse would

cause a momentum change of 720 kg•m/s. In a collision, the impulse experienced by an object is always equal to the momentum change.

Representing a Rebounding  Collision  Now consider a collision of a tennis ball with a wall. Depending on the physical  properties of the ball and wall, the speed at which the ball rebounds from the wall upon colliding with it will vary. The diagrams below depict the changes in velocity of the same  ball. For each representation (vector diagram, velocity-time graph, and ticker tape  pattern), indicate which case (A or B) has the greatest change in velocity, greatest acceleration, greatest momentum change, and greatest impulse.

Discussion

discussion concept  base quantities

translation

connection

s, r

s = θ × r

coordinate system r =  x  ̂ + y j ̂ s

velocity acceleration

v = d r/dt  a =

d v/dt  = d 2r/dt  2

v = ω × r

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r = r r ̂ + θ θ̂

ω = d θ/dt 

a = α × r − ω r

Translational and rotational quantities compared the museum of obsolete technology

θ

 x = r  cos θ  y = r  sin θ 2 2 r 2 =  x  + y −1 θ = tan  ( y/ x)

v0 + at  v =  x0 + v0t  + ½at  equations of moti  x = 2 on v2 = 2 v0  + 2a( x − x 0)

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rotation

2

α =

d ω/dt  = d 2θ/dt  2

0 + αt  ω = ω θ = θ2 0 + ω0t  + ½αt  ω2 = ω 2 + 2α(θ − θ 0

0)

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* period of rotation, rpm revolutions per minute, Hz = hertz = revolutions per second

Rotational Dynamics

Discussion

Taste and compare… translational

rotational

An object at rest tends to remain at rest and an object in motion tends to 1st continue moving  with constant velocity unless compelled by a net external force to act otherwise.

An object at rest tends to remain at rest and an object in rotation tends to continue rotating  with constant angular velocity unless compelled by a net external torque to act otherwise.

 Acceleration is directly proportional to net force and inversely proportional 2nd to massor ∑F = ma

angular acceleration is directly  proportional to net torque and inversely proportional to moment of inertia or ∑τ = I α

For every action there is an equal and 3rd opposite reaction. (Here action and reaction refer to forces.)

For every action there is an equal and opposite reaction. (Here action and reaction refer to torques.)

Translational and rotational laws of motion concept

translation

cause of acceleration resistance to acceleratio n newton's second law

connection

∑F

τ =

m

 I  =

∑F =

r × F

∑ r i2mi = ∫ r 2 dm

ma

rotation

∑τ  I 

∑ τ =  I α

Translational and rotational quantities compared Start with the equation for Newton's second law. Cross radius into both si des and then play around with the algebra. (This requires some unusual identities that I've seen before, but never really learned.)

∑F = ma

∑r × F = mr × a ∑r × F = mr × (α × r) ∑r × F = mα(r · r) − r(r · α) ∑r × F = mαr 2 − 0 ∑r × F = mr 2α ∑τ = α In the end, we get an analogous formula for Newton's second law of rotation with two new quantities: torque, the rotational equivalent to force τ = r × F and moment of inertia, the rotational equivalent to mass

⌠ ⌡

= ∑ r i2 mi = r 2 dm

Collisions A collision occurs when two or more objects hit each other. When objects collide, each object feels a force for a short amount of time. This force imparts an impulse, or changes the momentum of each of the colliding objects. But if the system of particles is isolated, we know that momentum is conserved. Therefore, while the momentum of each individual particle involved in the collision changes, the total momentum of the system remains constant. The procedure for analyzing a collision depends on whether the process is elastic or inelastic. Kinetic energy is conserved in elastic collisions, whereas kinetic energy is converted into other forms of energy during an inelastic collision. In both types of collisions, momentum is conserved. Elastic Collisions

Anyone who plays pool has observed elastic collisions. In fact, perhaps you’d better head over to the pool hall right now and start studying! Some kinetic energy is converted into sound energy when pool balls collide — otherwise, the collision would be silent — and a very small amount of kinetic energy is lost to friction. However, the dissipated energy is such a small fraction of the ball’s kinetic energy that we can treat the collision as elastic. Equations for Kinetic Energy and Linear Momentum

Let’s examine an elastic collision between two particles of mass

and , respectively. Assume that the collision is head-on, so we are dealing with only one dimension — you are unlikely to find two-dimensional collisions of any complexity on SAT II Physics. The velocities of the particles before the elastic collision are and , respectively. The velocities of the particles after the elastic collision are and . Applying the law of conservation of kinetic energy, we find:

Applying the law of conservation of linear momentum: These two equations put together will help you solve any problem involving elastic collisions. Usually, you will be given quantities for , , and , and can then manipulate the two equations to solve for and .

EXAMPLE

A pool player hits the eight ball, which is initially at rest, head-on with the cue ball. Both of these balls have the same mass, and the velocity of the cue  ball is initially . What are the velocities of the two balls after the collision? Assume the collision is perfectly elastic. Substituting we find:

and

into the equation for conservation of kinetic energy

Applying the same substitutions to the equation for conservation of momentum, we find:

If we square this second equation, we get: By subtracting the equation for kinetic energy from this equation, we get: The only way to account for this result is to conclude that and consequently . In plain English, the cue ball and the eight ball swap velocities: after the balls collide, the cue ball stops and the eight ball shoots forward with the initial velocity of the cue ball. This is the simplest form of an elastic collision, and also the most likely to be tested on SAT II Physics. Inelastic Collisions Most collisions are inelastic because kinetic energy is transferred to other forms of energy — such as thermal energy, potential energy, and sound — during the collision  process. If you are asked to determine if a collision is elastic or inelastic, calculate the kinetic energy of the bodies before and after the collision. If kinetic energy is not

conserved, then the collision is inelastic. Momentum is conserved in all inelastic collisions. On the whole, inelastic collisions will only appear on SAT II Physics qualitatively. You

may be asked to identify a collision as inelastic, but you won’t be expected to calculate

the resulting velocities of the objects involved in the collision. The one exception to this rule is in the case of completely inelastic collisions . Completely Inelastic Collisions

A completely inelastic collision, also called a “perfectly” or “totally” inelastic collision, is one in which the colliding objects stick together upon impact. As a result, the velocity of the two colliding objects is the same after they collide. Because , it is  possible to solve problems asking about the resulting velocities of objects in a completely inelastic collision using only the law of conservation of momentum. EXAMPLE

Two gumballs, of mass m and mass 2m respectively, collide head-on. Before impact, the gumball of mass m is moving with a velocity , and the gumball of mass 2m is stationary. What is the final velocity, , of the gumball wad? First, note that the gumball wad has a mass of m + 2m = 3m. The law of conservation of momentum tells us that , and so . Therefore, the final gumball wad moves in the same direction as the first gumball, but with one-third of its velocity. Collisions in Two Dimensions Two-dimensional collisions, while a little more involved than the one-dimensional

examples we’ve looked at so far, can be treated in exactly the same way as their one dimensional counterparts. Momentum is still conserved, as is kinetic energy in the case of elastic collisions. The significant difference is that you will have to break the trajectories

of objects down into x- and y-components. You will then be able to deal with the two components separately: momentum is conserved in the x direction, and momentum is conserved in the y direction. Solving a problem of two-dimensional collision is effectively the same thing as solving two problems of one-dimensional collision.

Because SAT II Physics generally steers clear of making you do too much math, it’s unlik ely that you’ll be faced with a problem where you need to calculate the final

velocities of two objects that collide two-dimensionally. However, questions that test your understanding of two-dimensional collisions qualitatively are perfectly fair game. EXAMPLE

A pool player hits the eight ball with the cue ball, as illustrated above. Both of the billiard balls have the same mass, and the eight ball is initially at rest. Which of the figures below illustrates a possible trajectory of the balls, given that the collision is elastic and both balls move at the same speed?

The correct answer choice is D, because momentum is not conserved in any of the other figures. Note that the initial momentum in the y direction is zero, so the momentum of the  balls in the y direction after the collision must also be zero. This is only true for choices D and E. We also know that the initial momentum in the xdirection is positive, so the final momentum in the x direction must also be positive, which is not true for E.

Friction

Frictional resistance to the relative motion of two solid objects is usually proportional to the force which presses the surfaces together as well as the roughness of the surfaces. Since it is the force perpendicular or "normal" to the surfaces which affects the frictional resistance, this force is typically called the "normal force" and designated by N. The frictional resistance force may then be written:

f friction = μN

μ = coefficient of friction μk  = coefficient of  kinetic friction μs = coefficient of  static friction

Standard model of friction

The frictional force is also presumed to be proportional to the coefficient of friction. However, the amount of force required to move an object starting from rest is usually greater than the force required to keep it moving at constant velocity once it is started. Therefore two coefficients of friction are sometimes quoted for a given  pair of surfaces - a coefficient of static friction and a coefficent of kinetic friction. The force expression above can be called the standard model of surface friction and is dependent upon several assumptions about friction. While this general description of friction (which I will refer to as the standard model) has practical utility, it is by no means a precise description of friction. Friction is in fact a very complex phenomenon which cannot be represented by a simple model. Almost every simple statement you make about friction can be countered with specific examples to the contrary. Saying that rougher surfaces experience more friction sounds safe enough - two pieces of coarse sandpaper will obviously be harder to move relative to each other than two pieces of fine sandpaper. But if two pieces of flat metal are made progressively smoother, you will reach a point where the resistance to relative movement increases. If you make them very flat and smooth, and remove all surface contaminants in a vacuum, the smooth flat surfaces will actually adhere to each other, making what is called a "cold weld".

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Index Friction concepts Plot of standard model Reference Jones & Childers, 2nd Ed. Sec 4.8

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Friction and Surface Roughness

Index Friction concepts Coefficients of friction

In general, the coefficients of friction for  static and kinetic friction are different. Like all simple statements about friction, this picture of friction is too simplistic. Saying that rougher surfaces experience more friction sounds safe enough - two pieces of coarse sandpaper will obviously be harder to move relative to each other than two pieces of fine sandpaper. But if two  pieces of flat metal are made progressively smoother, you will reach a  point where the resistance to relative movement increases. If you make them very flat and smooth, and remove all surface contaminants in a vacuum, the smooth flat surfaces will actually adhere to each other, making what is called a "cold weld". Once you reach a certain degree of mechanical smoothness, the frictional resistance is found to depend on the nature of the molecular forces in the area of contact, so that substances of

Plot of friction force Reference Jones & Childers, 2nd Ed. Sec 4.8

comparable "smoothness" can have significantly different coefficients of friction. An easily observed counterexample to the idea that rougher surfaces exhibit more friction is that of ground glass versus smooth glass. Smooth glass plates in contact exhibit much more frictional resistance to relative motion than the rougher ground glass.

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Index

Coefficients of Friction

Friction is typically characterized by a coefficient of friction which is the ratio of the frictional resistance force to the normal force which presses the surfaces together. In this case the normal force is the weight of the  block. Typically there is a significant difference between the coefficients of  static friction and kinetic friction.

Friction concepts Relation to surface roughness Plot of friction force

 Note that the static friction coefficient does not characterize static friction in general, but represents the conditions at the threshold of motion only. Friction and Automobile Tires

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