physics 1710 chapter 8—potential energy

Solution:Solution:

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

Power = dW/dt = (Fdx)/dt = F dx/dtPower = dW/dt = (Fdx)/dt = F dx/dt

= F v (for F constant)= F v (for F constant)

= (20.0 N )(36 x 10= (20.0 N )(36 x 1033 m/ 3600 sec) m/ 3600 sec)

= 200. N m/s = 200. W (#4)= 200. N m/s = 200. W (#4)

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

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

ReviewReview

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 - WK = U - W

½ mv ½ mv 22 + ½ ( + ½ (2/52/5 mv mv 22)= )=

mgh mgh

v v 22 = Rg = = Rg = 10/710/7 hg hg

h = 0.7 Rh = 0.7 R

h = h = 7/107/10(22.0 cm) = (22.0 cm) = 15.4 cm15.4 cmd = 2R+h = 69.4 cmd = 2R+h = 69.4 cm

vv

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

dd

Physics Works!Physics Works!

(When you include all relevant (When you include all relevant effects)effects)

11′′ Lecture Lecture

• Potential Energy is Potential Energy is U = -U = -∫ F•d r∫ F•d r

• The sum of all energy, potential and The sum of all energy, potential and kinetic, ofkinetic, of a system is conserved, in the absence of a system is conserved, in the absence of dissipation:dissipation:

E = U + K – WE = U + K – W

• F = - F = - ∇U = negative gradient of U.∇U = negative gradient of U.

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

Potential Energy:Potential Energy:

W = W = ∫ ∫ F•F•d d rr

U = -W = -∫ U = -W = -∫ F•F•d d rr

• 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.

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

What is the potential energy What is the potential energy of a 0.100 kg ball placed up a of a 0.100 kg ball placed up a 45 45 oo ramp 0.50 above the ramp 0.50 above the table?table?

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

FF

xx 0.50 0.50 mm

What is the potential energy What is the potential energy of a 0.100 kg ball placed up a of a 0.100 kg ball placed up a 45 45 oo ramp 0.50 above the ramp 0.50 above the table?table?

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

FF

xx 0.50 0.50 mm

- - mgmg

U = - FU = - F‧x = - (-‧x = - (-mg)hmg)h

= mg h= mg 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.

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

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

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

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 Chapter 8—Potential EnergyEnergy

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 ))

s= L s= L FFSS= - (1/L)dU/d = - (1/L)dU/d = - mg sin = - mg sin

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

ss

LL

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 ))

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

Example: Mass on a SpringExample: Mass on a Spring

Potential Energy:Potential Energy:

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.

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

Force:Force:

z = arz = ar22

U = mgzU = mgz

• FFrr = - = -∂U/∂r = ∂U/∂r = --∂(mgz)/∂r = -mg∂z/∂r ∂(mgz)/∂r = -mg∂z/∂r = - 2amgr = - k r= - 2amgr = - k r

Like a mass on a spring!Like a mass on a spring!

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

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

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

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

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

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

• F = - F = - ∇U∇U• 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

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy

Potential Energy:Potential Energy:

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

Physics 1710Physics 1710 Chapter 8—Potential Chapter 8—Potential EnergyEnergy