Matter & Interaction Chpt 6***

What are Work and Kinetic energy change proportional to?

discplacement

What are impulse and momentum change proportional to?

time interval

Energy Principle

Δ𝐸system+Δ𝐸surroundings=0

Gravitational Potential Energy

• Gravitational force 𝐹grav=−𝐺𝑚1𝑚2/𝑟^2 • Potential Energy 𝑈grav=−𝐺𝑚1𝑚2/𝑟 • 𝑟is the center -to -center separation of 𝑚1 and 𝑚2 • Near Earth: Near Earth: 𝑈grav = 𝑚𝑔𝑦

Work

•Amount of energy transfer as a force displaces an object•Between system and surroundings

Work and Energy

•Energy Principle: Difference Form o Change Esys = Wsurr + otherinputs •Energy Principle: Update Form o Esys,f = Esys,i + Wsurr + other inputs; Recall Esys= Erest + K

Sign of Work

•Force in the direction of motion: Work is > 0. •Force opposite to the direction of motion: Work is < 0. •Force perpendicular to the direction of motion: Work is = to 0. •Force that acts through zero displacement: Work is = to 0.

Negative Work:

•Parallel Force and Displacement are in opposite directions•Object slows down

Work

•Recall; Work=[Force]times [change in direction]=[Magnitude of Force] times [magnitude of direction] times [cosine theta]

Work Done by a Nonconstant Force

•Series of small displacements: Take limit as change of r0. •Replace change of rderivative r•Integrate from initial position to final position

Relationship Between Force and Potential Energy

𝐹𝑟=−𝑑𝑈/𝑑𝑟 • Force is the negative gradient of potential energy

**Work done by the friction depends on the path

--longer the path, the hotter the object gets.

Relationship between the relative directions of force and motion , and the sign of work done:

-A force in the direction of motion does positive work. -A force opposite to the direction of the motion does negative work. -A force perpendicular to the direction of the motion does zero work. -A force that acts through zero displacement does zero work.

What are the difference in the kinds of energy found in a single-particle system and a multiparticle system.

-A single-particle system has only particle energy[rest energy and kinetic energy] -a multiparticle system has particle energy and potential enregy[interaction energy]

Kinetic Energy at Low Speeds

-As the speed becomes low, you can use a binomial expansion of gamma(fig 6.19), gamma turns out to be 1, plus, one half of v, over, c squared, in the low speed limit and in that limit you can see then that the kinetic energy is just one-half, m v squared.

Why Energy?

-Can predict whether some processes are possible. -Give less detailed information. --No information about ---time ---change in direction

General Properties of Potential Energy

-Depends on particle separation, not on individual positions. • 𝑈→0 as 𝑟→∞ -Attractive interactions have negative potential energy as the distance decreases. -Repulsive interactions have positive potential energy as the distance decreases.

Path Independence

-How a particle moves from point A to Point B does not matter. -Same change in energy regardless of the path taken. -For a round trip, the change in potential energy must be zero.

--Work done by friction in Non-conservative forces;

-Interaction for which potential energy cannot be defined. -electric forces causes deformations of objects. -energy flows into vibration of the atoms.

Kinetic Energy

-Is the energy a particle has by virtue of its motion. -Purely as a consequence of its motion .

The Energy Principle is a fundamental principle because:

-It applies to every possible system, no matter how large or small it is, or how fast it is moving. It is true for any kind of interaction[gravitational, electromagnetic, strong, weak]. It relates an effect [change in energy of a system] to cause[an interaction witj the surroundings].

General Properties of Potential Energy

-Potential energy depends on the separation between pairs of particles, not on their individual positions. (As we saw with gravitational interactions, this depends on the reciprocity of gravitational and electric forces.) - Potential energy must approach zero as the separation between particles becomes very large, as we will soon show. -If an interaction is attractive, potential energy becomes negative as the distance between particles decreases. (We saw this in the case of gravitational interactions.) -If an interaction is repulsive, potential energy becomes positive as the distance between particles decreases.

The Energy Principle

-The change in the energy system is equal to the work done on the system by the surroundings plus the thermal energy transferred from the surroundings to the system. -The change in the energy of the system is going to be the change in all types of energy that are considered part of the system. -So remember, in any problem you've got to separate the system and the surroundings. -The energy is something you can calculate if you know the positions and velocities of all the particles in the system.

Potential Energy

-The kind of energy you need in order to take into account interactions between multiple particles in the system. -So if I include more than one particle in the system and they do work on one another, the potential energy keeps track of that work. -Therefore; the total energy is now going to become the total rest energy of all the particles plus the total kinetic energy of all the particles in the system, plus the potential energy the interaction between all the pairs of particles in the system.

Zero Work: Hockey puck sliding fig 6.11: -What objects are surrounding the puck? -Explain the work done by the force exerted by the Earth?

-There are three objects surrounding the puck: the hockey stick is pushing horizontally, doing positive work to speed up the puck; the Earth pulls down; and the ice pushes up, supporting the puck. -The work done by the Earth's gravitational is zero. Reason is, the force is perpendicular . A force perpendicular t the motion does zero work.

**Work Done by Friction

-When objects move with friction, their temperature increases -Temperature change indicates change in energy. -Work done by friction depends on the path--longer the path, the hotter the object gets. -Non-conservative force --interaction for which potential energy cannot be defined. --Electric forces cause deformation of objects. --Energy flows into vibration of the atoms.

***Work Done by Friction

-When objects move with friction, their temperatures increases. -Temperature change indicate change in energy. -Work done by the friction depends on the path--longer the path, the hotter the object gets. -Non-conservative forces; Interaction for which potential energy cannot be defined--electric forces causes deformations of objects--energy flows into vibration of the atoms.

Bound vs. Unbound States

-if K + U < 0, the system is in a bound state. -if K + U >= 0, the system is unbound -to find escape speed, let K + U =0 and solve for speed

Zero Work and Momentum: Fig 6.12, gravitational force of the earth on the moon; What kind of gravitational work is being done? What are the important differences between energy and momentum, and between work and impulse.

-the force is perpendicular to the motion, making the work zero. -there is no change in the moon's energy, since that depends on the magnitude of the velocity(constant), BUT there is a change in the moon's momentum, since that depends on the direction of the vector.

If you push or pull in the direction opposite to the displacement, you do _____ work and you ____ the energy of the system.

....you do negative work and you decrease the energy of the system.

If you push or pull in the direction of the displacement, you do ______ work and you ______ the energy of the system.

....you do positive work and you increase the energy of the system.

You drop a ball of mass m at a height h above the ground. The ball falls, speeding up, bounces off the floor, and goes upward, slowing down, until it is once again at the location where you released it (height h ). How much work was done by the Earth on the ball during this round trip?

0; No work done

Two ways to have Work = 0:

1) Change in direction is equal to zero; 2) Force and change in direction must be perpendicular

An isolated neutron decays : 𝑛→𝑝++𝑒−+ҧ 𝜈 Initial state: neutron at rest Final state: 𝑝+, 𝑒−, 𝜈far from each other Which statement is correct? 1. The sum of the rest energies of the products products equals the rest energy of the neutron. 2. The sum of the kinetic kinetic energies energies of the products equals equals the rest energy of the neutron . 3. The sum of the rest energies and kinetic energies of the products equals the rest energy of the neutron. 4. The sum of the kinetic energies of the products equals the kinetic energy of the neutron.

3. The sum of the rest energies and kinetic energies of the products equals the rest energy of the neutron.

What is the effect of increasing the displacement?

As in figure 6.6, if you lift a heavy box to a height above the ground, you will have expended a certain amount of chemical energy. If you lift the box as twice a high, you will have to have had to expend twice as much energy.

At low speeds how is the kinetic energy compared to the rest energy?

At low speeds, the kinetic energy is small compared to the rest energy. see fig 6.3.

To find the Work done on a system by the environment, you must know the external Force acting on the system and:

Displacement

If a particle is at rest, what is its energy?

Energy at rest is mass, times, speed of light squared: E = m c squared

The Energy Principle

Equals work plus thermal energy Work : -The surroundings can do work on the system and doing work changes the energy of the system. Thermal energy : You can also change the energy on the system by having thermal energy flow from the surroundings to the system, due to temperature differences between the system and the surroundings.

When Rest Energy does Not Change:

If a particle does NOT change its identity, both the initial and final energies of the system include rest energy[m c squared], and these cancel out in the energy equation.

Change of Rest Energy

If a particle doesn't change its identity • Mass doesn't change • Rest energy is constant • Process: Process: 𝐾𝑓=𝐾𝑖+𝑊 • Nuclear decay process • Example: Example: Example: 𝑛→𝑝^+ +𝑒^− + v

What is the effect of increasing the applied force?

If you increase the applied force, with a constant magnitude F in a constant direction, pulling through a displacement r. It will move move faster and faster, acquiring a certain amount of kinetic energy. Fig 6. 5.

What is the meaning of negative work?

If you want to slow down a moving object, you push in a direction opposite to the object's motion, and although the object keeps moving,in the original direction, it gradually slows down. The object's kinetic energy decreased, so you must have done negative work on the object.

When there is a Change in Rest Energy:

In neutron decay, a free neutron decays into a proton, an electron, and a nearly mass-less anti-neutrino, which travels nearly at the speed of light. All have kinetic energy. Therefore, this is a situation with a change of a particle identity, and a change in particle rest energy.

Work

Is something you calculate based on force and motion. If you have a force that acts while a particle is displaced by a distance displacement delta R, the work is simply the dot product of the force vector and the displacement vector. The dot product is defined as a way to multiply two vectors together, where you take the products of the corresponding components, and add the products together, to get the result in dot product.

Example: if you have a proton that is moving with a gamma of 4; What percentage of that energy will be kinetic energy?

Its energy is going to be four times its rest energy. Four times, mass[m], times, speed of light [c] squared. But seventy five percent of it is going to be kinetic energy, so, its kinetic energy is three times its rest energy. see fig 6. 2.

As a comet travels away from the star, how does the kinetic energy and potential energy of the system change?

Kinetic energy decreases, as the potential energy increases.

A skydiver falls toward the Earth, with his parachute open. During his fall, does the force by the air does positive, negative, or zero work on the system of skydiver plus parachute?

Negative

Along the ice, toward her partner. When she gets close he pushes on her in the +x direction, to slow her down. Does he do positive, negative ,or zero work?

Negative

The point of this question is to compare rest energy and kinetic energy at high speeds. An alpha particle (a helium nucleus) is moving at a speed of 0.9984 times the speed of light. Its mass is (6.40 10-27 kg). Is it okay to calculate its kinetic energy using the formula [1/2] mv^2?

No, because this formula isn't valid for speeds near the speed of light.

Is Doubling Displacement the same a s doubling the time?

No, it isn't. You are doing double the work, and twice as much kinetic energy. However, because the block is moving faster, in the second half of the double displacement, the time interval is less than double, so the momentum change is NOT doubled.

Positive Work:

Parallel Force and Displacement are in the same direction •Object speeds up

What are the differences between the definition of a particle energy and the definition of a particle momentum?

Particle energy is scalar; Particle momentum is a vector

A fancart moves in the -x direction.The fan is on, and the force on the cart by the air is also in the -x direction. Is the work done by the air positive, negative, or zero?

Positive

A tennis ball is moving in the -x direction. You hit it downward with a tennis racket. During the time your racket is in contact with the ball, do you do positive, negative, or zero work on the ball?

Positive

Solving Problems Involving Energy

Procedure: 1.Determine system and surroundings; 2.Identify the initial and final state of the system; 3.Write out all terms in the Energy Principle; .Identify terms that don't change & cancel; 5.Solve for unknown quantity; 6.Substitute numerical values you know

Single Particle vs. Multiparticle Systems

Single particle systems have: -Rest Energy -Kinetic Energy Multiparticle Systems have: -Rest energy -Kinetic energy -Potential energy

Identifying Initial and Final States

Some processes have several identifiable states -Example: Throwing a ball. --Initial height and velocity. --Turning point. --Final height and velocity.

Work and momentum: fig 6.13. Jumper. If the contact force by the floor does no work, in this scenario, what does this force do?

The force of the floor contributes to the net force that determines the change in the jumpers momentum, but the floor force does no work. The only significant work done by the surroundings is negative work doe by the force of the Earth(correspondingly, the chemical energy of the jumper decreases)

The point of this question is to compare rest energy and kinetic energy at high speeds. An alpha particle (a helium nucleus) is moving at a speed of 0.9984 times the speed of light. Its mass is (6.40 10-27 kg). Which is true? The kinetic energy is much smaller than the rest energy. The kinetic energy is much bigger than the rest energy. The kinetic energy is approximately equal to the rest energy

The kinetic energy is much bigger than the rest energy.

The point of this question is to compare rest energy and kinetic energy at low speeds. A baseball is moving at a speed of 34 m/s. Its mass is 145 grams (0.145 kg). -Which is true? The kinetic energy is much bigger than the rest energy. The kinetic energy is much smaller than the rest energy. The kinetic energy is approximately equal to the rest energy.

The kinetic energy is much smaller than the rest energy.

Energy of a Single Particle

The simplest case: -If you have a single particle and that is all there is in the system. -Everything else is in the surroundings -Then the energy is easy to compute -Energy is gamma, times mass(m), times speed of light squared If the particle is at REST; -it is mass(m), times speed of light© squared. If the particle is in MOTION; -Then in addition to the rest energy you get kinetic energy which is the energy particle has by virtue of the fact that it is moving and you can see that it just turns out to be gamma minus 1 times mc squared.

**A stationary neutron (n) decays into a proton (p + ), an electron (e ), and a nearly massless antineutrino (v) in a reaction that can be written n --> p^+ + e ^- + v. The sum of the ______________ of the products equals the ______________ of the neutron.

The sum of the REST ENERGIES AND KINETIC ENERGIES of the products equals the REST ENERGY of the neutron

A ball of mass 0.1 kg is dropped from rest near the Earth. The ball travels downward 2 m, speeding up. SYSTEM: Ball. Explain how you would obtain the work done by the surroundings?

To find the work done by the surroundings, we know that force is is in the negative y direction [force is mass, times gravity], and the displacement is also in the negative y direction, making the work positive. Work is = to force, times the displacement of y[both force and displacement are negative, making work positive]

A weight lifter picks up a barbell and 1. lifts it chest high 2. holds it for 30 seconds 3. puts it down slowly (but does not drop it). Rank the work W that the weight lifter does during each of these three operations. Label the quantities as W1, W2, and W3. (Hint: Think about how work is defined in terms of who is applying forces and who is doing work.)

W1>W2>W3 W1=force is in the direction of the motion, does positive work W2=force acts through zero displacement, does zero work W3=force opposite due to direction of motion, does negative work.

If a particle is moving with speed v, is its energy greater than m c squared?

Yes, If a particle is in motion, in addition to its rest energy, you get kinetic energy. Energy of the particle is = to the rest energy [m c squared], plus, the kinetic energy[gamma minus 1, times m c squared]

The point of this question is to compare rest energy and kinetic energy at low speeds. A baseball is moving at a speed of 34 m/s. Its mass is 145 grams (0.145 kg). -Is it okay to calculate its kinetic energy using the formula [1/2]mv^2?

Yes, because v, is < c.

Identifying Initial and Final States

Consider a nuclear reaction where -alpha particle collides with a carbon nucleus -produce an oxygen and nucleus with extra energy -oxygen nucleus emits excess energy as a photon. 1. Alpha and C moving while far apart 2. Alpha and C touch 3. Excited O* nucleus 4. O nucleus and photon

Electron Volt (eV) as a Unit of Energy

1eV=1.6×10−19J • Amount of energy an electron gains moving through electric potential difference of 1 volt Common rest energies: • Neutron: 939.6 MeV • Proton: 938.3 MeV • Electron: 0.511 MeV • Use rest energy to define mass: 𝐸rest=𝑚𝑐^2→𝑚=𝐸rest/𝑐^2 • Neutron: Neutron: Neutron: Neutron: 939.6MeV/𝑐^2

An electron (mass 9 10 31 kg) in a particle accelerator is acted on by a constant electric force⟨5 10 13 , 0, 0⟩N. (This is much greater than the force of gravity, which is only mg 1 10 29 N, so we can neglect the effect of the Earth on the electron.) The initial velocity of the electron is⟨0.995c, 0, 0⟩, where c as usual is the speed of light. The electron moves through a displacement of⟨5, 0, 0⟩m. (p230) 1. What is the system? 2. What is its surroundings? 3. What is the initial state? 4. What is the final state? 5. What can you find first to eventually solve for final velocity? 6. What equation do you need to use to extract this?

1. System: Electron. 2. Surroundings: Accelerator [which applies the electric force]; neglect the gravitational effect of the Earth in this short process. 3. Initial state: Beginning of given displacement. 4. Final state: End of given displacement. 5. It is easier to find gamma final to solve for final velocity. 6. We need to use the exact, relativistic equation for particle system.

**A stationary neutron (n) decays into a proton (p + ), an electron (e ), and a nearly mass-less anti-neutrino (v) in a reaction that can be written n --> p^+ + e ^- + v. 1. What is the system? 2. What is its surroundings? 3. What is the initial state? 4. What is the final state?

1. The system is the neutron. 2. The surroundings is space. It is not bound to a nucleus, making it unstable, therefore decaying. 3. Initial state is the neutron rest energy, plus the kinetic energy[which is zero]. 4. Final state is the sum of rest energies and the kinetic energies of the proton, electron, and anti-neutrino.

A ball of mass 0.1 kg is dropped from rest near the Earth. The ball travels downward 2 m, speeding up. SYSTEM: Ball, plus Earth. 1. What is the work done by the surroundings? 2. Did the kinetic energy of the ball + earth system change? 3. How about the work done by the internal forces?

1. The work done by the surroundings is zero, because there is not much in the surroundings exerting outside the system. 2. yes, it increased but not by much. However, this does not work unless we are neglecting another part of the system, potential energy[U], which is the interaction b/w the earth and the ball. 3. The work internal is also equal to the change in potential energy[U], therefore, the Work internal, which is equal to the change of potential energy had to decrease by the negative kinetic energy. **remember, potential energy is just bookeeping.

A ball is thrown straight up. Let the system be the ball, plus the earth. 1. What is the work done by the surroundings? 2. Explain the interacting energies in the system?

1. The work is zero, because there isn't much in the surroundings exerting in the system. 1. As the ball and the earth get closer together, the kinetic energy of the system increases, but we will see that the potential energy [interaction energy] decreases. The net change of the energy of the system is ZERO, which is consistent with the fact that no work was done on the system by the surroundings.

A ball is thrown straight up. Let the system be the ball. 1. What is the work done by the surroundings? 2. How is the kinetic energy of the system change?

1. Work is the force, time the displacement; force is the mass, times gravity[which is negative], times the displacement of the force, which is opposite of the direction of the force, making the work negative. 2. The kinetic energy of the system is the final kinetic energy of the ball at the height of the throw[where velocity is zero], minus the initial kinetic energy when the ball is thrown in the air. Since, final velocity is zero, you are left with the negative kinetic energy, which is less than zero[K < 0].

A horizontal spring has stiffness 100 N/m. A block is pressed against the spring, compressing the spring 0.2m, and then released. How would you determine how much work is done?

1. you would determine the initial and final states of the system. 2. Using the General definition of work, you would determine the work by dividing the compressing and releasing process into two stages. 3. to find the total work done, add both Work 1 and Work 2

Why cant a proton decay into a neutron?

Because the neutron has more rest energy than a proton. If a proton decays , the kinetic energy would be negative, A KINETIC ENERGY IS ALWAYS POSITIVE.

What are the similarities between the definition of a particle energy and the definition of a particle momentum?

Both contain the same factor gama; Both are proportional to the mass m.