Solution: Physics 1710 Chapter 7&8—Power & Energy K = ½ mv 2 = ½ (2.0 kg) (5.0 m/s) 2 = 25. kg m...

Solution:Solution:

Physics 1710Physics 1710 Chapter 7&8—Power & Chapter 7&8—Power & EnergyEnergy

K = ½ mv K = ½ mv 2 2

= ½ (2.0 kg) (5.0 m/s) = ½ (2.0 kg) (5.0 m/s) 22

= 25. kg m= 25. kg m22/s/s22

= 25. J= 25. J

What is the minimum height What is the minimum height from which a small rolling ball from which a small rolling ball must be started from rest so that must be started from rest so that it will complete a loop-the-loop?it will complete a loop-the-loop?


RR

hh

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What is the minimum height What is the minimum height from which a small rolling ball from which a small rolling ball must be started from rest so must be started from rest so that it will complete a loop-the-that it will complete a loop-the-loop?loop?


What is the minimum height What is the minimum height from which a small rolling ball from which a small rolling ball must be started from rest so that must be started from rest so that it will complete a loop-the-loop?it will complete a loop-the-loop?


RR

hh

vv22/R = g/R = g

K = U K = U

½ mv ½ mv 22 = mg(h-R) = mg(h-R)

v v 22 = 2 g (h-R) = 2 g (h-R)

g R = 2g (h-R)g R = 2g (h-R)

h = 3Rh = 3R

vv

What is the minimum height What is the minimum height from which a small rolling from which a small rolling ball must be started from ball must be started from rest so that it will complete a rest so that it will complete a loop-the-loop?loop-the-loop?


K = ½ mv K = ½ mv 2 2

= ½ (2.0 kg) (5.0 m/s) = ½ (2.0 kg) (5.0 m/s) 22

= 25. kg m= 25. kg m22/s/s22

= 25. J= 25. J

11′′ Lecture Lecture • Power is the time rate of change in Power is the time rate of change in energy. energy.

[Power]= [Watt] = [Joule]/[second][Power]= [Watt] = [Joule]/[second]

• Potential Energy Potential Energy UU is the energy stored is the energy stored in a in a system and may later produce work.system and may later produce work.

• The Potential Energy is equal to the The Potential Energy is equal to the negative of negative of the work done on the the work done on the system to put it in its system to put it in its present state.present state.

• The sum of all energy, potential and The sum of all energy, potential and kinetic, is kinetic, is conserved in an isolated system.conserved in an isolated system.


Power:Power:

P P = = dE/dtdE/dt

• Power is the time rate of change in the Power is the time rate of change in the energy of a system, the rate of work down energy of a system, the rate of work down on or by the system.on or by the system.

•Unit of power = Joule/second = WattUnit of power = Joule/second = Watt


Unit of Work and Energy:Unit of Work and Energy:

[[F F ‧ ‧ d d ] = N] = N‧m = Joule = ‧m = Joule = JJ

JouleJoule


Power: Power: Watt = Joule/secondWatt = Joule/second

James WattJames Watt

Watt’s Steam Engine Watt’s Steam Engine 17741774


Power:Power:

P P = = dE/dtdE/dt

P P = = ∆∆E/E/∆∆tt

∆∆E E = = PP ∆∆tt

∆∆E E ==(100 W)(3600 s) (100 W)(3600 s)

∆∆E E ==360 000 J = 360 kJ360 000 J = 360 kJ


Potential Energy:Potential Energy:

W = W = ∫ ∫ F•F•d d rr

U = -WU = -W

• Potential Energy is the negative of the work Potential Energy is the negative of the work required to put the system in the current state.required to put the system in the current state.




FF

-F-F

hh

U = U = - (- F h)- (- F h)

= = m g hm g h

Example: Elevated Mass Example: Elevated Mass

F = -mg F = -mg • Potential Energy:Potential Energy:

UUgg = - = -∫∫00

hhFdy = Fdy = --∫∫00

hh(- mg) dy(- mg) dy

UUgg = mg∫ = mg∫00

h h dy = mgh dy = mgh

• Thus, the potential energy stored in an elevated Thus, the potential energy stored in an elevated mass is proportional to the mass is proportional to the height hheight h and the and the weightweight of the mass.of the mass.


No Talking!No Talking!Think!Think!

Confer!Confer!

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Where does the energy come from to Where does the energy come from to produce electrical power in a produce electrical power in a hydroelectric dam?hydroelectric dam?




FF

-F-F

hh

U = U = m g hm g h

P P = = dU/dt dU/dt

= mg dh/dt= mg dh/dtmg =(100. mg =(100. kg)(9.8N/kg)kg)(9.8N/kg)

= 98.0 N= 98.0 N

dh/dt dh/dt = 10 m/10 = 10 m/10 s s

= 1 m/s= 1 m/s

P P = 98. W= 98. W

Relationship Between Relationship Between F F and and U:U:

U = -U = -∫ ∫ F•F•dd r r SoSo

U = -∫ [ FU = -∫ [ Fx x dx + Fdx + Fyy dy + F dy + Fzz dz] dz]ThenThen

FFx x =-dU/dx ; F=-dU/dx ; Fyy =-dU/dy; F =-dU/dy; Fzz =-dU/dz =-dU/dz

FF = - = -∇∇UU

F= -gradient of F= -gradient of UU


Example: Mass on a SpringExample: Mass on a Spring


U = ½ k x U = ½ k x 22

F =dU/dxF =dU/dxF= -½ k dxF= -½ k dx22/dx/dx

F= -k xF= -k x

• Thus, the force is equal to the negative of the gradient of Thus, the force is equal to the negative of the gradient of the potential energy.the potential energy.


The FThe Force is equal to the negative orce is equal to the negative gradient of the potential energy:gradient of the potential energy:

FF = - = -∇∇UUFFx x = -∂U/∂x = -∂U/∂x

FFyy = -∂U/∂y = -∂U/∂y

FFzz = -∂U/∂z = -∂U/∂z


Physics 1710Physics 1710 Chapter 8 Potential Energy and Chapter 8 Potential Energy and ConservationConservation

Example:Example: Ball on a slopeBall on a slope

• h = ax + byh = ax + by• U = mghU = mgh• FFxx = - = -∂U/∂x = ∂U/∂x = --∂(mgh)/∂x = -mg∂h/∂x ∂(mgh)/∂x = -mg∂h/∂x Similarly:Similarly:

FFyy = = --∂U/∂y = -mg b∂U/∂y = -mg b

• Thus, Thus, F = -F = -mgmg( ( aa i + i + b b j j ))

Conservation of EnergConservation of Energy:y:

• The sum of all energy in a system is The sum of all energy in a system is conserved, i.e. remains the same.conserved, i.e. remains the same.

E = U + KE = U + K


Example: PendulumExample: Pendulum

U = mg hU = mg hh = L(1- cos h = L(1- cos ))

U = mg L(1- cos U = mg L(1- cos ))

K = ½ m v K = ½ m v 22

=½ m (Ld =½ m (Ld /dt) /dt) 22

E = mg L(1- cos E = mg L(1- cos ) + ½ m (Ld ) + ½ m (Ld //dt) dt) 2 2

= constant= constant


Thought (Thought (Gedanken) Gedanken) Experiment:Experiment:

• Why does a pendulum stop Why does a pendulum stop moving?moving?


Dissipative (non-conservative) Forces:Dissipative (non-conservative) Forces:

W = W = ∫ ∫ F•F•d d rr

==∫ (C v∫ (C vxx 2 2 )dx)dx

==∫ (C v∫ (C vxx 2 2 )(dx /dt) dt)(dx /dt) dt

=∫ (C v=∫ (C vxx 3 3 )dt)dt

E = U + K -WE = U + K -W


Summary:Summary:

•The Potential Energy is equal to the The Potential Energy is equal to the negative of the work done on the system to negative of the work done on the system to put it in its present state.put it in its present state.

U = -U = -∫ F•d r∫ F•d r• The sum of all energy, potential and The sum of all energy, potential and kinetic, of a system is conserved, in the kinetic, of a system is conserved, in the absence of dissipation.absence of dissipation.

E = U + K – WE = U + K – W• F = - F = - ∇U∇U

•P P = = dE/dtdE/dt


Solution: Physics 1710 Chapter 7&8—Power & Energy K = ½ mv 2 = ½ (2.0 kg) (5.0 m/s) 2 = 25. kg m...

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Transcript of Solution: Physics 1710 Chapter 7&8—Power & Energy K = ½ mv 2 = ½ (2.0 kg) (5.0 m/s) 2 = 25. kg m...