Download LECTURE 27: Gravitational potential energy

LECTURE 27: Gravitational potential energy
Select LEARNING OBJECTIVES:
i. Construct an expression for the work due to gravity, defining this expression as gravitational
potential energy and identifying the work as conservative.
ii. Demonstrate the ability to extract forces from potential energy vs displacement graphs.
TEXTBOOK CHAPTERS:
• Giancoli (Physics Principles with Applications 7th) :: 6-4
• Knight (College Physics : A strategic approach 3rd) :: 10.4
• BoxSand :: Energy ( Potential Energy )
WARM UP: An non-constant force is applied to a box along the horizontal direction. Below is a graph of
the applied force verses horizontal position (x). What is the work done by this force on the box?
From the previous lecture, we see that work done on our system is equal to a change in kinetic energy of our system.
For this lecture, we will look at the work from the force of gravity acting on a system; here our system consists of
single object which is near Earth's surface.
EXAMPLE: Let's first consider a ball of mass m initially at some height (h) above the surface of the Earth. What is the
work that the force of gravity does on the ball as it falls to the surface?
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PRACTICE: A 2 kg object is initially at some height (h=5 meters) above the surface of the Earth as shown in the figure
below. The object rolls down the incline plane to the bottom. Consider a system consisting of just the 2 kg object.
What is the work that the force of gravity does on the ball as it rolls down the incline to the surface?
After walking away to take an important phone call, you come back to your favorite incline plane to find that
your engineering friends played a prank on you by changing the slope of the incline while keeping the height
at 5 meters. How much work does the force of gravity do on the 2 kg object starting from the top of the
incline to the bottom?
As seen in the practice problem above, the work from the force of gravity is the same regardless of which path the
object traversed. We can generalize further and claim that no matter what path, no matter how silly it may be, the work
from the force of gravity on an object is independent of the path the object went. This turns out to be very useful, so we
give this type of force a name, a conservative force. A conservative force is any type of force that when you calculate
the work due to this force on an object, the work is independent of the path the object travels. And while we are at it,
let's also call the work from the conservative forces "conservative work".
How is this helpful? Consider a system again consisting of a single object, perhaps a baseball tossed in the air like the
figure below. If we recognize that the force of gravity is a conservative force, we have already found the general form of
the work due to gravity near the Earth's surface, -mgΔy. So we do not have to do any vector operation to find out what
the work from gravity is, we just default to using the handy equation -mgΔy.
PRACTICE: Describe a possible scenario given the following information, the work due to gravity on an object is -20 J.
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Up to this point we have considered systems containing only one object, thus all the forces were external forces, so all
the work was external work. Now let's look at what happens when we have multiple objects contained in a system.
PRACTICE: Two massive objects are separated by some distance r as shown in the figure below. Consider a system that
contains both masses so that the force of gravity from each object on the other are internal forces. If the masses are
initially held at rest and then let go, what happens to the kinetic energy of the system?
1. Stays the same at 0 J.
2. Increases.
3. Decreases.
What is the sign of the net external work on the system?
(1) Positive
(2) Negative
(3) 0
Uh oh, what does this mean? How can the kinetic energy of the system be increasing but the net
external work on the system is zero? It turns out, when we are doing an energy analysis we need to be
careful and include the internal work due to internal forces. In the above example, the internal forces
were forces of gravity, thus the internal forces were also conservative forces. If there exists an internal
conservative force and if it is doing work within the system, "internal conservative work", then we
define what is called a potential energy. The important detail here is that we define potential energies
for internal conservative work, because the forces that are acting over a distance are conservative, we
do not need to directly calculate the work, rather we just use a derived potential energy function.
At this point, we need to revisit the work kinetic energy theorem which was stated as
But now we know better and we will include the net internal work and include the kinetic energy of
multiple objects in our system.
*Aside: Recall back to when we were looking at systems and doing a FBD analysis. If there were
internal forces we just ignored them, because if you summed the forces up, they cancel. For the above
example, the net force on the system is zero, which means that the acceleration of the center of mass of
the system is zero. Also momentum is conserved too since there are no external forces (the initial
momentum <0,0,> N s is the same as the final momentum <0,0> N s). Compare momentum and force
methods to an energy analysis, where we couldn't ignore the internal forces because the kinetic energy
of the system was increasing.
Gravitational potential energy (Ug)
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Gravitational potential energy (Ug)
Consider an object near the surface of the earth as shown in the figure below. Instead of defining our
system to be just the object, let's define our system to be the object+Earth.
The updated work kinetic energy equation for our system looks like…
First, notice that there is no forces crossing our system boundary, thus the net external work is zero, and
we can simplify the above work kinetic energy like so…
Now, we see that there is an internal force within our system, and it is not just any old internal force, it
is a conservative force as well. If there is an internal conservative force within our system, we define
what is called a potential energy. Here, the internal conservative force is the force of gravity, thus we
create what is known as the gravitational potential energy (Ug). This gravitational potential energy is the
system's (mass 1 + Earth) potential energy. Again, we now update our work kinetic energy equation…
Let' just algebraically rearrange the terms so that we get "final-initial" values so that we can better see
what is going on within our system…
So now we see that the gravitational potential energy of the system is being transformed into kinetic
energy of the system. This is our first energy transformation example. The negative change in
gravitational potential energy is there because of convention (the potential energy is smaller when the
objects are closer together, as they get closer to each other they are speeding up increasing kinetic
energy, thus we need the negative in front of the change in gravitational potential energy so that it is
also a positive value like the change in kinetic energy).
Now I need to clarify some approximations we often make. As the object falls towards the surface of
the Earth, the Earth also "falls" towards the object. However, since the Earth is much more massive
than the object, we can approximate that the Earth basically does not move at all. So as the ball falls the
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than the object, we can approximate that the Earth basically does not move at all. So as the ball falls the
potential energy of the system (Earth + ball) decreases, but since the Earth basically doesn't move, we
often just say that the "ball's" gravitational potential energy decreases instead of the "ball + Earth"
system's gravitational potential energy decreases. Likewise, if the ball is raised up to some height above
the Earth's surface, we say that the "ball's" potential energy increases, instead of the "ball + Earth"
system's potential energy increases. With this approximation we would write our energy equation for
our system (mass 1 + Earth) as the following…
Since we said the Earth basically doesn't move, that means the velocity of the Earth does not change, so
the change in kinetic energy of the Earth is zero and the terms cancel out. Also, since the Earth basically
does not move, the force of gravity from mass 1 on the Earth does not act over any distance, so the
gravitational potential energy of the system is often just referred to as the gravitational potential energy
of the mass 1. So in the end, we have an energy equation with terms only referring to our mass 1, but
our system was the combined (mass 1 + Earth) system. Recall back to when we defined a system of just
an object near the surface of the Earth, we found that the force of gravity did conservative work equal
to -mgΔy. Now that we have made our approximations that the Earth does not move, the gravitational
potential energy of an object near the Earth is…
Notice how our energy equation is always in terms of initial and final quantities, so for completeness the
following is how we write the initial, final, and the change in gravitational potential energy. The
convention is to define the positive y-direction to be upwards.
We can now also relate the conservative internal work due to gravity to the change in gravitational
potential energy…
*Side note: In general, for any internal conservative work, we can define a potential energy…
For example, we will talk about spring potential energy soon, and electric potential energy much later.
PRACTICE: A box slides down an incline as shown below. Use an the work kinetic energy analysis to find
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PRACTICE: A box slides down an incline as shown below. Use an the work kinetic energy analysis to find
the final velocity of the box at the bottom of the incline.
PRACTICE: Consider 2 objects, both near the surface of Earth as shown in the figure below. No other
forces are acting on each object other than the force of gravity from the Earth. If object 1 is moving
upward, and object 2 is moving downward, what is the energy equation if your system is (Earth + object
1 + object 2).
PRACTICE: An object near the surface of the earth falls from some height h to the surface. Draw a
physical representation of this scenario. Also sketch a graph representing gravitational potential energy
vs position.
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Questions for discussion:
(1) Why is the work due to gravity written as
?
(2) One of your friends claims that the gravitation potential energy of a book on a shelf is 25 J.
Another friend claims that it is 50 J. Is one of your friends necessarily wrong?
(3) As a box slides down an incline, is the change in gravitational potential energy dependent on the
angle of the incline? Is the work from gravity dependent on the angle of the incline?
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