Download y -y =h

An object of mass m
moves from point A to
point B
yB-yA=h
y
yB
yA
B
A
xA
xB
x
Completely independent of the path
The only thing that matters is the difference in height
Whenever the work that is done by a force is independent of its path-it's only determined by the starting point and the end point-- displacement
that force is called a "conservative force." It's a very important concept in physics.
Gravity is a conservative force. 1
An object of mass m
moves from point A to
point B
yB-yA=h
y
yB
yA
B
A
xA
xB
x
Work energy theorem mgy is called gravitational potential energy denoted by U
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The sum of potential energy at point B and the kinetic energy at point B is the same as the potential energy at A and the kinetic energy at point A. One can be converted into the other and it can be converted back. Kinetic energy can be converted back to potential energy, and potential energy can be converted back, but the sum of them‐which we call "mechanical energy“
is conserved.
Mechanical energy is only conserved if the force is a conservative force.
Amazing result and a very useful tool !
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Conservation of Mechanical Energy
Definition of mechanical energy:
Using this definition and considering only
conservative forces, we find:
Or equivalently:
The Work Done by Conservative Forces
Gravity example
If we pick up a ball and put it on the shelf, we
have done positive work on the ball.
We can get that energy back if the ball falls
back off the shelf; in the meantime, we say
the energy is stored as potential energy.
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Clicker
Two paths lead to the top of a big hill. One is steep and direct, while the other is twice as long but less steep. How much more potential energy would you gain if you take the longer path?
a) the same
b) twice as much
c) four times as much
d) half as much
e) you gain no PE in either case
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Initial Potential Energy
Initial total energy
energy is conserved
energy is conserved
energy is conserved
Final total energy
Final =Initial
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Clicker
a
b
c
d) same speed
for all ramps
Three balls of equal mass start from rest and roll down different ramps. All ramps have the same height. Which ball has the greater speed at the bottom of its ramp?
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A, vA=0
D, vD
h
B, vB
R
2R
y
y
x
Point B:
Point D:
y=0, U=0
C, vc=max velocity
let y be a variable for every point
(1)
However there is an addition condition for point D for the ball to stay on the track
(2)
(1)
Substitute
equation (2)
for vD
Min criteria
Solve for h
d
Assuming m2 to be larger, the
system will accelerate in the
direction indicated.
Since the acceleration is
same for both masses,
they can be treated as a
system with total mass
m2+m1
Net force on the two
mass system is the
difference in the
masses
Equation of motion
can be written as
Kinematic equation
From chapter 2
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Conservation of Energy
Potential Energy terms
Kinetic Energy terms
d
s
d
Put everything together
Same solution using Newton’s 2nd law
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