Chapter 8Chapter 8Conservation of EnergyConservation of Energy

Types of SystemsTypes of Systems

Non-isolated systemsNon-isolated systems Energy can cross the system boundary in a variety of ways.Energy can cross the system boundary in a variety of ways. Results in a Total energy of the system changesResults in a Total energy of the system changes

Isolated systemsIsolated systems Energy does not cross the boundary of the systemEnergy does not cross the boundary of the system Total energy of the system is constantTotal energy of the system is constant

Conservation of energyConservation of energy Can be used if no non-conservative forces act within the Can be used if no non-conservative forces act within the

isolated systemisolated system Applies to biological organisms, technological systems, Applies to biological organisms, technological systems,

engineering situations, etcengineering situations, etc

Introduction

Examples of Ways to Examples of Ways to Transfer EnergyTransfer Energy

Section 8.1

Conservation of EnergyConservation of Energy

Energy is conservedEnergy is conserved This means that energy cannot be created nor This means that energy cannot be created nor

destroyed.destroyed. If the total amount of energy in a system changes, it If the total amount of energy in a system changes, it

can only be due to the fact that energy has crossed can only be due to the fact that energy has crossed the boundary of the system by some method of the boundary of the system by some method of energy transfer.energy transfer.

Section 8.1

Conservation of EnergyConservation of Energy

Mathematically, Mathematically, EEsystemsystem = = EEsystemsystem is the total energy of the system is the total energy of the system TT is the energy transferred across the system boundary is the energy transferred across the system boundary by by

some mechanism some mechanism Established symbols: Established symbols: TTworkwork = = WW and and TTheatheat = = QQ Others just use subscriptsOthers just use subscripts

The primarily mathematical representation of the energy The primarily mathematical representation of the energy version of the analysis model of the non-isolated system is version of the analysis model of the non-isolated system is given by the full expansion of the above equation.given by the full expansion of the above equation.

K + K + U +U +EEintint = W + Q + T = W + Q + TMWMW + T + TMTMT + T + TETET + T + TERER TTMWMW – transfer by mechanical waves – transfer by mechanical waves TTMTMT – by matter transfer – by matter transfer TTETET – by electrical transmission – by electrical transmission TTER ER – by electromagnetic transmission – by electromagnetic transmission

Section 8.1

Conservation of EnergyConservation of Energy

Isolated SystemIsolated System

For an isolated system, For an isolated system, EEmechmech = 0 = 0

Remember ERemember Emechmech = K + U = K + U This is This is conservation of conservation of

energyenergy for an isolated for an isolated system with no non-system with no non-conservative forces conservative forces acting.acting.

Conservation of Energy Conservation of Energy becomes becomes EEsystemsystem = 0 = 0

EEsystemsystem is all kinetic, is all kinetic, potential, and internal potential, and internal energiesenergies

Example – Ball in Free FallExample – Ball in Free Fall

A ball of mass m is dropped from a A ball of mass m is dropped from a height h above the ground as shown in the height h above the ground as shown in the figure to the right. figure to the right.

Neglecting air resistance, determine the Neglecting air resistance, determine the speed of the ball when it is at a height y speed of the ball when it is at a height y above the ground. Choose the system as above the ground. Choose the system as the ball and the Earth. the ball and the Earth.

ExampleExampleThe spring is compressed The spring is compressed to position A, and the trigger to position A, and the trigger is fired. The projectile of is fired. The projectile of mass, m, rises to a position C mass, m, rises to a position C above the position at which it above the position at which it leaves the spring (shown in leaves the spring (shown in the figure as position B, where the figure as position B, where y =0) . If the mass is 35.0 y =0) . If the mass is 35.0 grams, A = -0.120 m, and C = grams, A = -0.120 m, and C = 20.0 m, and neglecting all 20.0 m, and neglecting all resistive forces, determine the resistive forces, determine the spring constant. spring constant.

Find the speed of the Find the speed of the projectile as it moves through projectile as it moves through the equilibrium position B of the equilibrium position B of the spring. the spring.

Kinetic FrictionKinetic Friction

Kinetic friction can be modeled Kinetic friction can be modeled as the interaction between as the interaction between identical teeth.identical teeth.

The frictional force is spread out The frictional force is spread out over the entire contact surface.over the entire contact surface.

The displacement of the point of The displacement of the point of application of the frictional force is application of the frictional force is not calculable.not calculable.

Therefore, the work done by the Therefore, the work done by the frictional force is not calculable.frictional force is not calculable.

Section 8.3

Work and Energy With FrictionWork and Energy With Friction

In general, if friction is acting in a system:In general, if friction is acting in a system: KK = = WWother forcesother forces - -ƒƒkkdd This is a modified form of the work – kinetic energy theorem.This is a modified form of the work – kinetic energy theorem.

Use this form when friction acts on an object.Use this form when friction acts on an object. If friction is zero, this equation becomes the same as Conservation of If friction is zero, this equation becomes the same as Conservation of

Mechanical Energy.Mechanical Energy.

A friction force transforms kinetic energy in a system to internal energy.A friction force transforms kinetic energy in a system to internal energy.

The increase in internal energy of the system is equal to its decrease in The increase in internal energy of the system is equal to its decrease in kinetic energy.kinetic energy.

EEintint = ƒ = ƒkk d d

In general, this equation can be written as In general, this equation can be written as ΣΣWWother forcesother forces = W = = W = ΔΔK + K + ΔΔEEint int

This represents the non-isolated system model for a system within which This represents the non-isolated system model for a system within which a non-conservative force acts.a non-conservative force acts.

Section 8.3

ExampleExample

A 6.0 kg block initially at rest A 6.0 kg block initially at rest is pulled to the right along a is pulled to the right along a horizontal surface by a horizontal surface by a constant horizontal force of 12 constant horizontal force of 12 N. Find the speed of the block N. Find the speed of the block after it has moved 3.0 m if the after it has moved 3.0 m if the surfaces in contact have a surfaces in contact have a coefficient of kinetic friction of coefficient of kinetic friction of 0.15. 0.15.

Suppose the force is applied Suppose the force is applied at an angle as shown in b. At at an angle as shown in b. At what angle should the force what angle should the force be applied to achieve the be applied to achieve the largest possible speed after largest possible speed after the block has moved 3.0 m to the block has moved 3.0 m to the right? the right?

Section 8.3

ExampleExample

A block of mass 1.6 kg is A block of mass 1.6 kg is attached to a horizontal spring that attached to a horizontal spring that has a force constant of 1000 N/m has a force constant of 1000 N/m as shown. The spring is as shown. The spring is compressed 2.0 cm and is then compressed 2.0 cm and is then released from rest. released from rest.

Calculate the speed of the block Calculate the speed of the block as it passes through the as it passes through the equilibrium position x =0 if the equilibrium position x =0 if the surface is frictionless. surface is frictionless.

Calculate the speed of the block Calculate the speed of the block as it passes through the as it passes through the equilibrium position if a constant equilibrium position if a constant friction force of 4.0 N retards its friction force of 4.0 N retards its motion from the moment it is motion from the moment it is released. released.

Section 8.3

Adding Changes in Adding Changes in Potential EnergyPotential Energy

If friction acts within an isolated systemIf friction acts within an isolated system

EEmechmech = = K + K + U = -ƒU = -ƒkk d dU is the change in all forms of potential energyU is the change in all forms of potential energy

If non-conservative forces act within a non-isolated system and the external If non-conservative forces act within a non-isolated system and the external influence on the system is by means of work.influence on the system is by means of work.

EEmechmech = -ƒ = -ƒkk d + d + WWother forcesother forces

This equation represents the non-isolated system model for a system that This equation represents the non-isolated system model for a system that possesses potential energy and within which a non-conservative force acts possesses potential energy and within which a non-conservative force acts and can be rewritten asand can be rewritten as

ΣΣWWother forcesother forces = W = = W = ΔΔK + K + ΔΔU + U + ΔΔEEint int

Section 8.4

ExampleExample

A 3.00 kg crate slides down a A 3.00 kg crate slides down a ramp. The ramp is 1.00 m in ramp. The ramp is 1.00 m in length and inclined at an angle length and inclined at an angle 30.0 degrees as shown. The crate 30.0 degrees as shown. The crate starts from rest at the top, starts from rest at the top, experiences a constant friction experiences a constant friction force of magnitude 5.00 N, and force of magnitude 5.00 N, and continues to move a short continues to move a short distance on the horizontal floor distance on the horizontal floor after it leaves the ramp. after it leaves the ramp.

Use the energy methods to Use the energy methods to determine the speed of the crate determine the speed of the crate at the bottom of the ramp. at the bottom of the ramp.

How far does the crate slide on How far does the crate slide on the horizontal floor if it continues the horizontal floor if it continues to experience a friction force of to experience a friction force of magnitude 5.00 N? magnitude 5.00 N?

Section 8.4

ExampleExample

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

If friction is present, the energy If friction is present, the energy decreases.decreases.

EEmechmech = - = -ƒƒkkdd

A block having a mass of 0.80 kg is A block having a mass of 0.80 kg is given an initial velocity va = 1.2 m/s given an initial velocity va = 1.2 m/s to the right and collides with a to the right and collides with a spring whose mass is negligible spring whose mass is negligible and whose force constant is k = 50 and whose force constant is k = 50 N/m. Assuming the surface to be N/m. Assuming the surface to be frictionless, calculate the maximum frictionless, calculate the maximum compression of the spring after the compression of the spring after the collision. collision. Section 8.4

ExampleExample

The block of mass mThe block of mass m11 lies on a lies on a horizontal surface and is horizontal surface and is connected to a spring of force connected to a spring of force constant k. The system is constant k. The system is released from rest when the released from rest when the spring is unstretched. If the spring is unstretched. If the hanging block of mass mhanging block of mass m22 falls a falls a distance h before coming to rest, distance h before coming to rest, calculate the coefficient of kinetic calculate the coefficient of kinetic friction between the block mass friction between the block mass mm11 and the surface. and the surface.

ExampleExample

Section 8.4

PowerPower

Power is the time rate of energy transfer.Power is the time rate of energy transfer.

The The instantaneous powerinstantaneous power is defined as is defined as

Using work as the energy transfer method, this can Using work as the energy transfer method, this can also be written asalso be written as

Section 8.5

Instantaneous Power and Instantaneous Power and Average PowerAverage Power

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

This expression for power is valid for any means of This expression for power is valid for any means of energy transfer.energy transfer.

Section 8.5

Units of PowerUnits of Power

The SI unit of power is called the watt.The SI unit of power is called the watt. 1 watt = 1 joule / second = 1 kg 1 watt = 1 joule / second = 1 kg .. m m22 / s / s33

A unit of power in the US Customary system is A unit of power in the US Customary system is horsepower.horsepower.

1 hp = 746 W1 hp = 746 W

Units of power can also be used to express units of Units of power can also be used to express units of work or energy.work or energy.

1 kWh = (1000 W)(3600 s) = 3.6 x101 kWh = (1000 W)(3600 s) = 3.6 x1066 J J

Section 8.5

ExampleExample

An elevator car has a mass of 1600 kg and is An elevator car has a mass of 1600 kg and is carrying passengers having a combined mass carrying passengers having a combined mass of 200 kg. A constant friction force of 4000 N of 200 kg. A constant friction force of 4000 N retards its motion. How much power must a retards its motion. How much power must a motor deliver to lift the elevator car and its motor deliver to lift the elevator car and its passengers at a constant speed of 3.00 m/s?passengers at a constant speed of 3.00 m/s?

Problem Solving Summary Problem Solving Summary – Non-isolated System– Non-isolated System

The most general The most general statement describing the statement describing the behavior of a non-isolated behavior of a non-isolated system is the conservation system is the conservation of energy equation.of energy equation.

ΔΔEEsystemsystem = = ΣΣTT

This equation can be This equation can be expanded or have terms expanded or have terms deleted depending upon deleted depending upon the specific situation.the specific situation.

Summary

Problem Solving Summary Problem Solving Summary – Isolated System– Isolated System

The total energy of an The total energy of an isolated system is conservedisolated system is conserved

ΔΔEEsystemsystem = 0 = 0

If no non-conservative If no non-conservative forces act within the isolated forces act within the isolated system, the mechanical system, the mechanical energy of the system is energy of the system is conserved.conserved.

ΔΔEEmechmech = 0 = 0Summary