Transcript Gravitational potential energy

© Simon Porter 2007
Gravitational Potential Energy
How much GPE?
How much GPE?
GPE = mgh?
How much GPE?
GPE = mgh?
How much GPE?
GPE = mgh?
How much GPE?
GPE = mgh?
We do know that the
GPE must be
decreasing. But where is
the GPE zero?
How much GPE?
GPE = mgh?
We do a little physicists
trick. We take the GPE at
infinity to be zero! That
means that it has negative
GPE at distance closer than
infinity!
© Simon Porter 2007
Gravitational potential energy
Gravitational potential energy at a point
is defined as the work done to move a
mass from infinity to that point.
© Simon Porter 2007
Gravitational potential energy
Gravitational potential energy at a point
is defined as the work done to move a
mass from infinity to that point.
I’ve come from
infinity!
m
M
R
© Simon Porter 2007
Gravitational potential energy
Gravitational potential energy at a point
is defined as the work done to move a
mass from infinity to that point.
Work done = force x
distance
I’ve come from
infinity!
The force however is
changing as the
mass gets closer
m
M
R
© Simon Porter 2007
Gravitational potential energy
R
R
R
W=
Fdr
=
GMmdr
r2
=
[ ]
- GMm
GMm
=
r
R
I’ve come from
infinity!
m
M
R
© Simon Porter 2007
Gravitational potential energy
Gravitational potential energy at a point
is defined as the work done to move a
mass from infinity to that point.
Ep = -GMm
r
Ep is always negative
© Simon Porter 2007
Gravitational Potential
It follows that the Gravitational potential
at a point is the work done per unit mass
on a small point mass moving from infinity
to that point. It is given by
Ep = mV
V = -GM
r
Note the difference between gravitational potential energy (J) and
Gravitational potential (J.kg-1)
Moving masses in potentials
If a mass is moved from a position with
potential V1 to a position with potential V2,
work = m(V2 – V1) = mΔV
V2
(independent of path)
V1
Equipotential surfaces/lines
Equipotential surfaces/lines
Field and equipotentials
• Equipotentials are always perpendicular to field lines.
Diagrams of equipotential lines give us information about
the gravitational field in much the same way as contour
maps give us information about geographical heights.
Field strength = potential gradient
In fact it can be shown from calculus that
the gravitational field is given by the
potential gradient (the closer the
equipotential lines are together, the
stronger the field)
g = -dV
dr
Let’s stop and read!
Pages 127 to 130
Pages142 to 151
© Simon Porter 2007
Escape speed
Imagine throwing a ball into the air
© Simon Porter 2007
Escape speed
It falls to the ground (Doh!)
© Simon Porter 2007
Escape speed
What happens if you throw harder?
© Simon Porter 2007
Escape speed
It goes higher and takes longer to return.
© Simon Porter 2007
Escape speed
It goes higher and takes longer to return.
Ouch!
© Simon Porter 2007
Escape speed
The kinetic energy of the ball changes to
gravitational potential energy as the ball
rises. This in turn turns back into kinetic
energy as the ball falls again.
© Simon Porter 2007
Escape speed
How fast would you have to throw the ball
so that it doesn’t come back? (i.e. goes to
“infinity” or escapes the gravitational field
of the earth)
© Simon Porter 2007
Escape speed
At “infinity”, it gravitational energy is given
by Ep = -GMm/r
= zero when r is infinite
© Simon Porter 2007
Escape speed
Energy conservation tells us that it must
therefore have zero energy to start with if it
is to escape the earth’s gravity.
i.e. KE + GPE = 0
© Simon Porter 2007
Escape speed
i.e. KE + GPE = 0
½mv2 + -GMem/Re = 0
(where Re is the radius of the earth)
½mv2 = GMem/Re
v = √2GMe/Re
© Simon Porter 2007
Escape speed
v = √2GM/Re
v = √(2 x 6.67 x 10-11 x 5.98 x 1024)/6.38 x 106
v = 12000 m.s-1
I can’t throw that
fast!
In reality the
escape
velocity of the
earth is bigger
than this.
WHY?
Let’s try some questions!
Page 153 Q7, 13
© Simon Porter 2007
Hold on!
Isn’t electricity
similiar?
© Simon Porter 2007
Gravitational Potential
The Gravitational potential at a point is
the work done per unit mass on a small
point mass moving from infinity to that
point. It is given by
Ep = mV
V = -GM
r
Note the difference between gravitational potential energy (J) and
Gravitational potential (J.kg-1)
© Simon Porter 2007
Electrical Potential
The Electrical potential at a point is the
work done per unit charge on a small
positive test charge moving from infinity to
that point. It is given by
Uel = qV
Scalar quantity
V=W
q
Note the difference between electrical potential energy (J) and Electrical
potential (J.C-1)
Moving charges in potentials
If a charge is moved from a position with
potential V1 to a position with potential V2,
work = q(V2 – V1) = qΔV
V2
(independent of path)
V1
© Simon Porter 2007
Gravitational potential energy
Gravitational potential energy at a point
is defined as the work done to move a
mass from infinity to that point.
Ep = -GMm
r
Ep is always negative
© Simon Porter 2007
Electrical potential energy
Electrical potential energy at a point is
defined as the work done to move a
positive charge from infinity to that point.
Uel = kQq
r
Equipotential surfaces/lines
Ep = -GMm
r
Equipotential surfaces/lines
Field and equipotentials
• Equipotentials are always perpendicular to field lines.
Diagrams of equipotential lines give us information about
the gravitational field in much the same way as contour
maps give us information about geographical heights.
Field strength = potential gradient
In fact it can be shown from calculus that
the gravitational field is given by the
potential gradient (the closer the
equipotential lines are together, the
stronger the field)
E = dV
dr
© Simon Porter 2007
From “Physics for the IB Diploma”
K.A.Tsokos (Cambridge University Press)
Gravitation
Electricity
Acts on
Mass (always +?)
Charge (+ or -)
Force
F = GM1M2/r2
Attractive only, infinite
range
F = kQ1Q2/r2
Attractive or repulsive,
infinite range
Relative strength
1
1042
Field
g = GM/r2
E = kQ/r2
Potential
V = -GM/r
V = kQ/r
Potential energy
Ep = -GMm/r
Ep = kQq/r
Let’s try some questions
Pages 307 Questions
2, 4, 5, 6, 11, 12.