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Chapter 7. Potential Energy. 7.1 Potential Energy. Potential energy is the energy associated with the configuration of a system of objects that exert forces on each other The forces are internal to the system. Types of Potential Energy. There are many forms of potential energy, including:

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Chapter 7

Chapter 7

Potential Energy


7 1 potential energy
7.1 Potential Energy

  • Potential energy is the energy associated with the configuration of a system of objects that exert forces on each other

    • The forces are internal to the system


Types of potential energy
Types of Potential Energy

  • There are many forms of potential energy, including:

    • Gravitational

    • Electromagnetic

    • Chemical

    • Nuclear

  • One form of energy in a system can be converted into another


System example
System Example

  • 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

Fig 7.1


Potential energy
Potential Energy

  • The energy storage mechanism is called potential energy

  • A potential energy can only be associated with specific types of forces

  • Potential energy is always associated with a system of two or more interacting objects


Gravitational potential energy
Gravitational Potential Energy

  • Gravitational Potential Energy is associated with an object at a given distance above Earth’s surface

  • Assume the object is in equilibrium and moving at constant velocity

  • The work done on the object is done by and the upward displacement is


Gravitational potential energy cont
Gravitational Potential Energy, cont

  • The quantity mgy is identified as the gravitational potential energy, Ug

    • Ug = mgy

  • Units are joules (J)


Gravitational potential energy final
Gravitational Potential Energy, final

  • The gravitational potential energy depends only on the vertical height of the object above Earth’s surface

  • In solving problems, you must choose a reference configuration for which the gravitational potential energy is set equal to some reference value, normally zero

    • The choice is arbitrary because you normally need the difference in potential energy, which is independent of the choice of reference configuration


7 2 conservation of mechanical energy
7.2 Conservation of Mechanical Energy

  • The mechanical energy of a system is the algebraic sum of the kinetic and potential energies in the system

    • Emech = K + Ug

  • The statement of Conservation of Mechanical Energy for an isolated system is Kf + Ugf= Ki+ Ugi

    • An isolated system is one for which there are no energy transfers across the boundary


Conservation of mechanical energy example
Conservation of Mechanical Energy, example

  • Look at the work done by the book as it falls from some height to a lower height

  • Won book = DKbook

  • Also, W = mgyb – mgya

  • So, DK = -DUg

Fig 7.2


Chapter 7 4247690

Active Figure

7.3

NEXT


Chapter 7 4247690

A ball of mass m is dropped from rest at a height h above the ground as in Figure 7.4. Ignore air resistance.

Fig 7.4


Chapter 7 4247690

You are designing apparatus to support an actor ofmass 65 kg who is to “fly” down to the stage during the performance of a play. You attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us identify the angle as the angle that the actor’s cable makes with the vertical when he begins his motion from rest. What is the maximum value  can have such that the sandbag does not lift off the floor during the actor’s swing?


Fig 7 5
Fig 7.5 who is to “fly” down to the stage during the performance of a play. You attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us identify the angle


Chapter 7 4247690

Since who is to “fly” down to the stage during the performance of a play. You attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us identify the angle T = mbagg


7 3 conservative forces
7.3 Conservative Forces who is to “fly” down to the stage during the performance of a play. You attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us identify the angle

  • A conservative force is a force between members of a system that causes no transformation of mechanical energy within the system

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

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


Nonconservative forces
Nonconservative Forces who is to “fly” down to the stage during the performance of a play. You attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us identify the angle

  • A nonconservative force does not satisfy the conditions of conservative forces

  • Nonconservative forces acting in a system cause a change in the mechanical energy of the system


Nonconservative force example
Nonconservative Force, Example who is to “fly” down to the stage during the performance of a play. You attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us identify the angle

  • 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

Fig 7.7


Elastic potential energy
Elastic Potential Energy who is to “fly” down to the stage during the performance of a play. You attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us identify the angle

  • Elastic Potential Energy is associated with a spring, Us = 1/2 k x2

  • The work done by an external applied force on a spring-block system is

    • W = 1/2 kxf2 – 1/2 kxi2

    • The work is equal to the difference between the initial and final values of an expression related to the configuration of the system


Elastic potential energy cont
Elastic Potential Energy, cont who is to “fly” down to the stage during the performance of a play. You attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us identify the angle

  • This expression is the elastic potential energy: Us = 1/2 kx2

  • The elastic potential energy can be thought of as the energy stored in the deformed spring

  • The stored potential energy can be converted into kinetic energy

Fig 7.6


Elastic potential energy final
Elastic Potential Energy, final who is to “fly” down to the stage during the performance of a play. You attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us identify the angle

  • The elastic potential energy stored in a spring is zero whenever the spring is not deformed (U = 0 when x = 0)

    • The energy is stored in the spring only when the spring is stretched or compressed

  • The elastic potential energy is a maximum when the spring has reached its maximum extension or compression

  • The elastic potential energy is always positive

    • x2 will always be positive


Chapter 7 4247690

Active Figure who is to “fly” down to the stage during the performance of a play. You attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us identify the angle

7.6

NEXT


Conservation of energy extended
Conservation of Energy, Extended who is to “fly” down to the stage during the performance of a play. You attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us identify the angle

  • Including all the types of energy discussed so far, Conservation of Energy can be expressed as

    DK + DU + DEint = DE system = 0 or

    K + U + E int = constant

    • K would include all objects

    • U would be all types of potential energy


Problem solving strategy conservation of mechanical energy
Problem Solving Strategy – Conservation of Mechanical Energy

  • Conceptualize: Define the isolated system and the initial and final configuration of the system

    • The system may include two or more interacting particles

    • The system may also include springs or other structures in which elastic potential energy can be stored

    • Also include all components of the system that exert forces on each other


Problem solving strategy 2
Problem-Solving Strategy, 2 Energy

  • Categorize: Determine if any energy transfers across the boundary

    • If it does, use the nonisolated system model, DE system = ST

    • If not, use the isolated system model, DEsystem = 0

  • Determine if any nonconservative forces are involved


Problem solving strategy 3
Problem-Solving Strategy, 3 Energy

  • Analyze: Identify the configuration for zero potential energy

    • Include both gravitational and elastic potential energies

    • If more than one force is acting within the system, write an expression for the potential energy associated with each force


Problem solving strategy 4
Problem-Solving Strategy, 4 Energy

  • If friction or air resistance is present, mechanical energy of the system is not conserved

  • Use energy with non-conservative forces instead

    • The difference between initial and final energies equals the energy transformed to or from internal energy by the nonconservative forces


Problem solving strategy 5
Problem-Solving Strategy, 5 Energy

  • If the mechanical energy of the system is conserved, write the total energy as

    • Ei = Ki + Ui for the initial configuration

    • Ef = Kf + Uf for the final configuration

  • Since mechanical energy is conserved, Ei = Ef and you can solve for the unknown quantity


Mechanical energy and nonconservative forces
Mechanical Energy and Nonconservative Forces Energy

  • In general, if friction is acting in a system:

    • DEmech = DK + DU = -ƒkd

    • DU is the change in all forms of potential energy

    • If friction is zero, this equation becomes the same as Conservation of Mechanical Energy


Chapter 7 4247690

A 3.00-kg crate slides down a ramp at a loading dock. The ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

Fig 7.8


Nonconservative forces example 1 slide
Nonconservative Forces, Example 1 (Slide) ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

DEmech = DK + DU

DEmech =(Kf – Ki) +

(Uf – Ui)

DEmech = (Kf + Uf) –

(Ki + Ui)

DEmech = 1/2 mvf2 – mgh = -ƒkd


Nonconservative forces example 2 spring mass
Nonconservative Forces, Example 2 (Spring-Mass) ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

  • Without friction, the energy continues to be transformed between kinetic and elastic potential energies and the total energy remains the same

  • If friction is present, the energy decreases

    • DEmech = -ƒkd


7 4 conservative forces and potential energy
7.4 Conservative Forces and Potential Energy ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

  • 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


Conservative forces and potential energy
Conservative Forces and Potential Energy ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

  • 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


Conservative forces and potential energy check
Conservative Forces and Potential Energy – Check ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

  • 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


7 5 nonisolated system in steady state
7.5 Nonisolated System in Steady State ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

  • A system can be nonisolated with 0 = DT

    • This occurs if the rate at which energy is entering the system is equal to the rate at which it is leaving

    • There can be multiple competing transfers


Nonisolated system in steady state house example
Nonisolated System in Steady State, House Example ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

Fig 7.11


7 6 potential energy for gravitational forces
7.6 Potential Energy for Gravitational Forces ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

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

  • The potential energy then is

Fig 7.12


Potential energy for gravitational forces final
Potential Energy for Gravitational Forces, Final ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

  • The result for the earth-object system can be extended to any two objects:

  • For three or more particles, this becomes


Fig 7 13
Fig 7.13 ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.


Fig 7 14
Fig 7.14 ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.


Electric potential energy
Electric Potential Energy ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

  • Coulomb’s Law gives the electrostatic force between two particles

  • This gives an electric potential energy function of


7 7 energy diagrams and stable equilibrium
7.7 Energy Diagrams and Stable Equilibrium ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

  • 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

Fig 7.15


Energy diagrams and unstable equilibrium
Energy Diagrams and Unstable Equilibrium ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

  • 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

Fig 7.16


Chapter 7 4247690

Active Figure ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

7.15

NEXT


Chapter 7 4247690

Active Figure ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

7.15b

NEXT


Neutral equilibrium
Neutral Equilibrium ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

  • 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


7 8 potential energy in fuels
7.8 Potential Energy in Fuels ramp is 1.00 m in length and is inclined at an angle of 30.0° as shown in Figure 7.8. The crate starts from rest at the top and experiences a constant friction force of magnitude 5.00 N. Use energy methods to determine the speed of the crate when it reaches the bottom of the ramp.

  • Fuel represents a storage mechanism for potential energy

  • Chemical reactions release energy that is used to operate the automobile