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### Chapter 7

Kinetic energy and work

Kinetic energy K is energy associated with the state of motion of an object. The faster the object moves, the greater is its kinetic energy.

For an object of mass m whose speed v is well below the speed of light,

The SI unit of kinetic energy (and every other type of energy) is the joule (J),

1 joule = 1 J = 1 kgm2/s2.

Work W is energy transferred to or from an object by means of a force acting on the object.

Energy transferred to the object is positive work, and energy transferred from the object is negative work.

To calculate the work a force F does on an object as the object moves through some displacement d, we use only the force component along the object’s displacement. The force component perpendicular to the displacement direction does zero work.

For a constant force F, the work done W is:

A constant force directed at angle f to the displacement (in the x-direction) of a bead does work on the bead. The only component of force taken into account here is the x-component.

When two or more forces act on an object, the net work done on the object is the sum of the works done by the individual forces.

7.6: Work done by gravitational force

(b) An applied force lowers an object. The displacement of the object makes an angle with the gravitational force .The applied force does negative work on the object.

(a) An applied force lifts an object. The object’s displacement makes an angle f =180° with the gravitational force on the object. The applied force does positive work on the object.

Work-kinetic energy theorem

ΣW = Kf – Ki= ΔK

It holds for both positive and negative work: If the net work done on a particle is positive, then the particle’s kinetic energy increases by the amount of the work, and the converse is also true.

The theorem says that the change in kinetic energy of a particle is the net work done on the particle.

Workdone onanaccelerating elevator cab

7.8: Work done by a general variable force

A. One-dimensional force, graphical analysis:

- We can divide the area under the curve of F(x) into a number of narrow strips of width x.
- We choose x small enough to permit us to take the force F(x) as being reasonably constant over that interval.
- We let Fj,avg be the average value of F(x) within the jth interval.
- The work done by the force in the jth interval is approximately
- I
- Wj is then equal to the area of the jth rectangular, shaded strip.

7.8: Work done by a general variable force

A. One-dimensional force, calculus analysis:

We can make the approximation better by reducing the strip width Dx and using more strips (Fig. c). In the limit, the strip width approaches zero, the number of strips then becomes infinitely large and we have, as an exact result,

7.8: Work done by a general variable force

B. Three dimensional force:

If

where Fx is the x-components of F and so on,

and

where dx is the x-component of the displacement vector drand so on,

then

Finally,

Hooke’s Law

When x is positive (spring is stretched), F is negative

When x is 0 (at the equilibrium position), F is 0

When x is negative (spring is compressed), F is positive

7.7: Work done by a spring force

Fs = - kx

Slide 11

Work Done by a Spring

Identify the block as the system

Calculate the work as the block moves from xi = -xmax to xf = 0

The total work done as the block moves from

–xmax to xmax is zero

xf

xi

Slide 12

7.7: Work done by a spring force

Hooke’s Law: To a good approximation for many springs, the force from a spring is proportional to the displacement of the free end from its position when the spring is in the relaxed state. The spring force is given by

The minus sign indicates that the direction of the spring force is always opposite the direction of the displacement of the spring’s free end. The constant k is called the spring constant (or force constant) and is a measure of the stiffness of the spring.

The net work Ws done by a spring, when it has a distortion from xi to xf , is:

Work Ws is positive if the block ends up closer to the relaxed position (x =0) than it was initially. It is negative if the block ends up farther away from x =0. It is zero if the block ends up at the same distance from x= 0.

Work Done by Gravitational Force

Generalizing gravitational potential energy uses Newton’s Law of Universal Gravitation:

Calculate the work as the objectmoves from ri to rf

Slide 14

7.8: Work kinetic energy theorem with a variable force

A particle of mass m is moving along an x axis and acted on by a net force F(x) that is directed along that axis.

The work done on the particle by this force as the particle moves from position xi to position xfis :

But,

Therefore,

Instantaneous Power

The instantaneous power is the limiting value of the average power as Dt approaches zero

This can also be written as

The SI unit of power is the joule per second, or Watt (W).

In the British system, the unit of power is the footpound

per second. Often the horsepower is used.

1N=1 kg-m/s2

= 0.225 lb

1 m = 3.281 ft

Slide 17

Potential energy and conservation of energy

8.1 Potential energy

- Technically, potential energy is energy that can be associated with the configuration (arrangement) of a system of objects that exert forces on one another.
- Some forms of potential energy:
- Gravitational Potential Energy,
- Elastic Potential Energy

Conservative Forces

The work done by a conservative force on a particle moving between any two points isindependent of the path taken by the particle

The work done by a conservative force on a particle moving through any closed path is zero

Slide 21

8.3 Path Independence of Conservative Forces

The net work done by a conservative force on a particle moving around any closed path is zero.

If the work done from a to b along path 1 as Wab,1 and the work done from b back to a along path 2 as Wba,2. If the force is conservative, then the net work done during the round trip must be zero

If the force is conservative,

Nonconservative Force, Example

Friction is an example of a nonconservative force

The work done depends on the path

The red path will take more work than the blue path

Slide 23

This system consists of Earth and a book

Do work on the system by lifting the book through Dy

The work done is mgyb - mgya

Determining Potential Energy

Gravitational Potential Energy

Ub

F外力

Ua

= mgyb- mgya

Slide 24

8.4: Determining Potential Energy values:

For the most general case, in which the force may vary with position, we may write the work W:

8.4: Determining Potential Energy values:

Elastic Potential Energy

In a block–spring system, the block is moving on the end of a spring of spring constant k. As the block moves from point xi to point xf , the spring force Fx =- kx does work on the block. The corresponding change in the elastic potential energy of the block–spring system is

If the reference configuration is when the spring is at its relaxed length,

and the block is at xi = 0.

Find W for each one of the three paths.

8.5: Conservation of Mechanical Energy

Principle of conservation of energy:

In an isolated system where onlyconservative forces cause energy changes, the

kinetic energy and potential energy can change, but their sum, the mechanical energy Emec of the system, cannot change.

The mechanical energy Emec of a system is the sum of its potential energy U and the kinetic energy K of the objects within it:

With and

We have:

8.7: Work done on a System by an External Force

Work is energy transferred to or from a system by means of an external force acting on that system.

(b) Once the rescue is complete, Tarzan and Jane must swing back across the river. With what minimum speed must they begin their swing ?

(a) With what minimum speed must Jane begin her swing to just make it to the other side ?

mJ

mT

Fig. P7-57, p.218

8.8: Conservation of Energy back across the river. With what minimum speed must they begin their swing ?

Law of Conservation of Energy

The total energy E of a system can change only by amounts of energy that are transferred to or from the system.

where Emec is any change in the mechanical energy of the system, Eth is any change in the thermal energy of the system, and Eint is any change in any other type of internal energy of the system.

The total energy E of an isolated system cannot change.

8.8: Conservation of Energy back across the river. With what minimum speed must they begin their swing ?

External Forces and Internal Energy Transfers

internal energy=biochemical in the muscles

An external force can change the kinetic energy or potential energy of an object without doing work on the object—that is, without transferring energy to the object. Instead, the force is responsible for transfers of energy from one type to another inside the object.

F back across the river. With what minimum speed must they begin their swing ?net=Σfriction ( f )

The net external force Fnet from the road change the kinetic energy of the car. However, wiFnetdoes not transfer energyfrom the roadto the carand so out does no work on the car.Instead, the force is responsible for transfers of energy from the energy stored in the fuel.

Conservative Forces and Potential Energy back across the river. With what minimum speed must they begin their swing ?

Define a potential energy function, U, such that the work done by a conservative force equals the decrease in the potential energy of the system

The work done by such a force, F, is

DU is negative when F and x are in the same direction

Slide 35

Conservative Forces and Potential Energy back across the river. With what minimum speed must they begin their swing ?

The conservative force is related to the potential energy function through

The conservative force acting between parts of a system equals the negative of the derivative of the potential energy associated with that system

This can be extended to three dimensions

Slide 36

Conservative Forces and Potential Energy – Check back across the river. With what minimum speed must they begin their swing ?

Look at the case of an object located some distance y above some reference point:

This is the expression for the vertical component of the gravitational force

Slide 37

Energy Diagrams and Stable Equilibrium back across the river. With what minimum speed must they begin their swing ?

The x = 0 position is one of stable equilibrium

Configurations of stable equilibrium correspond to those for which U(x) is a minimum

x=xmax and x=-xmax are called the turning points

Slide 38

Energy Diagrams and Unstable Equilibrium back across the river. With what minimum speed must they begin their swing ?

Fx = 0 at x = 0, so the particle is in equilibrium

For any other value of x, the particle moves away from the equilibrium position

This is an example of unstable equilibrium

Configurations of unstable equilibrium correspond to those for which U(x) is a maximum

Slide 39

Solution back across the river. With what minimum speed must they begin their swing ?

- Stable equilibrium exists for a separation distance at which the potential energy of the system of two atoms (the molecule) is a minimum. Take the derivative of the function U(x):

Slide 41

Solution back across the river. With what minimum speed must they begin their swing ?

- Minimize the function U(x) by setting its derivative equal to zero:
- Evaluate xeq the equilibrium separation of the two atoms in the molecule:

Slide 42

Solution back across the river. With what minimum speed must they begin their swing ?

- We graph the Lennard-Jones function on both sides of this critical value to create our energy diagram as shown in Figure.

Slide 43

Solution back across the river. With what minimum speed must they begin their swing ?

- Notice that U(x) is extremely large when the atoms are very close together, is a minimum when the atoms are at their critical separation, and then increases again as the atoms move apart. When U(x) is a minimum, the atoms are in stable equilibrium. indicating that the most likely separation between them occurs at this point.

Slide 44

Neutral Equilibrium back across the river. With what minimum speed must they begin their swing ?

Neutral equilibrium occurs in a configuration when U is constant over some region

A small displacement from a position in this region will produce neither restoring nor disrupting forces

Slide 45

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