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Physics 1710 Chapter 7&8—Power & Energy
Solution:
K = ½ mv 2
= ½ (2.0 kg) (5.0 m/s) 2
= 25. kg m2/s2
= 25. J
Physics 1710 Chapter 7&8—Power & Energy
What is the minimum height from which a
small rolling ball must be started from rest
so that it will complete a loop-the-loop?
h
R
Physics 1710 Chapter 7&8—Power & Energy
What is the minimum height from which a
small rolling ball must be started from rest
so that it will complete a loop-the-loop?
Think!
Peer Instruction Time
No Talking!
Confer!
Physics 1710 Chapter 7&8—Power & Energy
What is the minimum height from which a
small rolling ball must be started from rest
so that it will complete a loop-the-loop?
v2/R = g
K=U
½ mv 2 = mg(h-R)
v
v 2 = 2 g (h-R)
g R = 2g (h-R)
h = 3R
R
h
Physics 1710 Chapter 7&8—Power & Energy
What is the minimum height from
which a small rolling ball must be
started from rest so that it will
complete a loop-the-loop?
K = ½ mv 2
= ½ (2.0 kg) (5.0 m/s) 2
= 25. kg m2/s2
= 25. J
Physics 1710 Chapter 7&8—Power & Energy
1′ Lecture
• Power is the time rate of change in energy.
[Power]= [Watt] = [Joule]/[second]
• Potential Energy U is the energy stored in a
system and may later produce work.
• The Potential Energy is equal to the negative of
the work done on the system to put it in its
present state.
• The sum of all energy, potential and kinetic, is
conserved in an isolated system.
Physics 1710 Chapter 7&8—Power & Energy
Power:
P = dE/dt
Power is the time rate of change in the energy
of a system, the rate of work down on or by the
system.
•
•Unit of power = Joule/second = Watt
Physics 1710 Chapter 7&8—Power & Energy
Unit of Work and Energy:
[F ‧ d ] = N‧m = Joule = J
Joule
Physics 1710 Chapter 7&8—Power & Energy
Power: Watt = Joule/second
James Watt
Watt’s Steam Engine 1774
Physics 1710 Chapter 7&8—Power & Energy
Power:
P = dE/dt
P
= ∆E/∆t
∆E = P ∆t
∆E
=(100 W)(3600 s)
∆E
=360 000 J = 360 kJ
Physics 1710 Chapter 7&8—Power & Energy
Potential Energy:
W = ∫ F•d r
U = -W
• Potential Energy is the negative of the work required to
put the system in the current state.
Physics 1710 Chapter 7&8—Power & Energy
Potential Energy:
-F
F
U=
h
=
- (- F h)
mgh
Physics 1710 Chapter 7&8—Power & Energy
Example: Elevated Mass
F = -mg
• Potential Energy:
h
h
Ug = -∫0 Fdy = -∫0 (- mg) dy
h
Ug = mg∫0 dy = mgh
• Thus, the potential energy stored in an elevated mass is
proportional to the height h and the weight of the mass.
Physics 1710 Chapter 7&8—Power & Energy
Where does the energy come from to produce
electrical power in a hydroelectric dam?
Think!
Peer Instruction Time
No Talking!
Confer!
Physics 1710 Chapter 7&8—Power & Energy
Potential Energy:
-F
F
h
U=
mgh
P
= dU/dt
= mg dh/dt
mg =(100. kg)(9.8N/kg)
= 98.0 N
dh/dt = 10 m/10 s
= 1 m/s
P
= 98. W
Physics 1710 Chapter 7&8—Power & Energy
Relationship Between F and U:
So
Then
U = -∫ F•d r
U = -∫ [ Fx dx + Fy dy + Fz dz]
Fx =-dU/dx ; Fy =-dU/dy; Fz =-dU/dz
F = -∇U
F= -gradient of U
Physics 1710 Chapter 7&8—Power & Energy
Example: Mass on a Spring
Potential Energy:
U=½ kx2
F =dU/dx
F= -½ k dx2/dx
F= -k x
• Thus, the force is equal to the negative of the gradient of the
potential energy.
Physics 1710 Chapter 7&8—Power & Energy
The Force is equal to the negative gradient of
the potential energy:
F = -∇U
Fx = -∂U/∂x
Fy = -∂U/∂y
Fz = -∂U/∂z
Physics 1710
Chapter 8 Potential Energy and Conservation
Example: Ball on a slope
• h = ax + by
• U = mgh
• Fx = -∂U/∂x = -∂(mgh)/∂x = -mg∂h/∂x
Similarly:
Fy = -∂U/∂y = -mg b
• Thus, F = -mg( a i + b j )
Physics 1710 Chapter 7&8—Power & Energy
Conservation of Energy:
• The sum of all energy in a system is conserved,
i.e. remains the same.
E=U+K
Physics 1710 Chapter 7&8—Power & Energy
Example:
Pendulum
U = mg h
h = L(1- cos  )
U = mg L(1- cos  )
K=½mv2
=½ m (Ld /dt) 2
E = mg L(1- cos  ) + ½ m (Ld /dt) 2
= constant
Physics 1710 Chapter 7&8—Power & Energy
Thought (Gedanken) Experiment:
• Why
does a pendulum stop
moving?
Physics 1710 Chapter 7&8—Power & Energy
Dissipative (non-conservative) Forces:
W = ∫ F•d r
=∫ (C vx 2 )dx
=∫ (C vx 2 )(dx /dt) dt
=∫ (C vx 3 )dt
E = U + K -W
Physics 1710 Chapter 7&8—Power & Energy
Summary:
•The Potential Energy is equal to the negative of
the work done on the system to put it in its
present state.
U = -∫ F•d r
• The sum of all energy, potential and kinetic, of a
system is conserved, in the absence of
dissipation.
E=U+K–W
• F = - ∇U
•P = dE/dt