Download Gravity and Orbits Lesson - The Ohio State University

Lesson Plan
Lesson Title
How Gravity Affects Orbits
Grade Level
State Indicator(s)
Goals and Objectives Students will explore how the force of gravity affects celestial bodies to
for Student Learning cause one to orbit around the other (i.e., mimicking the planets orbiting the
sun in the solar system).
Main points to be covered include:
1.) Gravity is always an attractive force operating between any two
masses
2.) The strength of the force depends directly on the masses of the two
bodies.
3.) The strength of the force depends inversely on the distance between
the two objects.
4.) Whether an object is captured into a gravitational orbit depends on
the mass, radius, and velocity of the approaching object.
Diversity
Teaching Method
Learning Activities
Guided to open inquiry
Science Introduction:
Gravity is an attractive force that operates between ALL objects that have
mass. The mathematical equation for the force of gravity is:
F = GMm
d2
where M and m are the masses of the two objects on which the gravitational
force is acting upon, d is the distance between the two objects, and G is the
gravitational constant, measured to be 6.67 x 10-11 m3 kg-1 s-2 through
experiments (i.e., empirically).
In non-math speak, this equation means that ALL pairs of objects both feel
the gravitational force from other objects AND exert a gravitational force on
other objects (i.e., Newton’s 3rd Law: For every force, there is an equal and
opposite force). So that the gravitational force acting between two objects
is actually felt by both objects. The strength of this force is ONLY
dependent on the masses of both objects AND the distance between them, in
the sense that the force becomes STRONGER for LARGER masses, and the
force becomes WEAKER with LARGER distances between the massive
objects.
Since gravity is an attractive force between all massive objects, both on
earth and in space, it is responsible for the planets staying in orbit around
the sun. If gravity were to suddenly stop acting between the sun and the
planets in the solar system, the planets would just fly off in straight lines,
going which ever direction they happened to be moving at the moment the
gravity stopped. However, since the gravity attracts the planets and the sun,
this force constantly pulls the planets toward the sun, even when they are
trying to move in a straight line (i.e., Newton’s 1st Law: An object in
motion stays in motion unless acted upon by an outside force), resulting in
the orbital motion that we see.
Lesson Introduction:
In this lesson, pairs of students will experiment with different strengths and
lengths of yarn while moving in marked paths around the room to simulate
the force of gravity. They will experiments with the string to see if they can
deduce the dependencies of gravity: directly dependent on mass, inversely
dependent on distance.
Demonstration:
Example of Newton’s 1st Law: in the absence of gravity (an outside force),
objects in orbit will begin to travel in straight lines.
Tie a donut to a string (cake donuts work best), and swing it around in a
circle in the air above your head. As it swings, the string will cut into the
donut, and when the string eventually cuts all the way through the donut, it
is released from the force directing it in a circle (due to the string), and will
fly off in some direction in a straight line. (This can also be demonstrated
by simply letting go of the string – with anything attached to the other end –
but this isn’t quite as dramatic)
Main Procedure:
0.) Mark an X on the floor with masking tape, and then next to the X,
lines along the floor 2 feet, 4 feet, and 8 feet away, e.g. ,black marks
below:
Sun
Planet
2 ft
START
4 ft
START
8 ft
START
1.) Provide each student with several 8 ft. lengths (~10) of yarn.
2.) The general procedure will involve one student standing on the ‘X’
(i.e., the sun) holding one end of the string. The other student(s)
will begin quickly walking (or a slow jog) along the lines (depicted
above) near the person on the ‘X’. As the student walking the line
crosses in front of the person on the ‘X’, the force of gravity is
enacted, another student standing on the red line hands the string to
the walking student (alternately, rig something up so the student can
grab the yarn as they walk by so they don’t have to stop), and
‘attempts’ to continue walking the straight line. If the force of
gravity is strong enough, the student’s path will be altered, and they
will be forced to ‘orbit’ the person on the ‘X’, moving off of the
straight marked path. However, for the weakest cases, the student
will most likely just break the string and keep moving on their way –
the force of gravity wasn’t strong enough to ‘capture’ the student
into orbit. Teacher note: Please make it clear to students that they
should not be “jerking” on the string to try to break it, as adding this
extra force to “try” to break the string will inevitably make it more
difficult for them to easily notice the trends in the strength of the
‘gravitational force’ (or string).
3.) Cases to try:
a.) Variable Mass: Have a single student walk the same line
over and over, but vary the mass of the ‘sun’ by adding more
students = more lengths of string to the ‘X’ position, so that
each time, as the student starts to move past on the line, they
first pick up 1 string from one student, two strings from two
students, three strings from three students, etc. For each
case, the strength of the gravitational force should become
greater because the mass is increased, so the “string” (now
multiple strings) holding the two bodies together is harder to
break too.
b.) Variable distance: Have student walk each of the three lines
at varying distance from the ‘X’, only this time, they have to
use the same length of string each time, which means that
they have to double up and then quadruple the 8ft length for
the two closer lines to walk (4ft and 2 ft, respectively). In
this scenario, the strength of the force between two bodies
gets stronger with closer distances, even though the mass in
the system is not changing. (Alternately, if you wish them to
start on the closest line and see the force getting weaker with
distance, then have them start with the quadrupled length of
string and gradually unfold it as they move to the farther
lines). Teachers Note: This doesn’t accurately represent the
inverse square nature of the force (e.g., the force is 4 times
weaker if the distance between the objects is doubled) but
will at least give the general inverse dependence of the
Materials
Supplements
gravitational force on distance.
4.) Have students record their experiences, and discuss and even graph
their results.
5.) To make this more open inquiry, teachers don’t have to give the
students the gravitational force equation or any preconceived ideas
of what they should get. Intuition should tell students that more
lengths of string will be stronger and more difficult to “break free”
from than a single length, which should help them understand the
cases in which the force is stronger, though you may have to point
this out in the discussion.
Skein of yarn (cheap “Red Heart” brand polyester works well)
Masking tape
Yard Stick
Cake donut (for demo.)
Escape Velocity:
In order for an object to escape the gravitational pull of another object, the
escaping object must be traveling at a high enough velocity, called the
“escape velocity,” or else the attractive force will be greater and will pull
the objects back together. Similarly, in order for one object to be captured
into orbit in the gravitational field of a larger object (like the sun capturing a
planet), the planet must pass close enough and slow enough by the sun.
This is where the saying “what goes up, must come down” comes from. It
comes back down because it wasn’t going up fast enough to escape the
attractive force of gravity pulling it back down. Given that it is possible to
escape an object’s gravitational pull, the saying should really be “What goes
up too slowly, must come down.”
You can derive the escape velocity using conservation of energy. In the
scenario of an object escaping the gravitational force of another object, it
must travel an infinitely large distance away, and to calculate the exact
velocity or speed it needs to make its escape, by the time it reaches its final
destination (at infinite distance), it’s final velocity will be zero. So to
conserve energy, you must have the initial energy equal the final energy
(both kinetic and potential), where the kinetic energy of an object is given
by KE = 1/2mv2 and the gravitational potential energy is PE = -GMm/r (it’s
negative because the gravitational force resulting the potential energy is
attractive and so said to be negative). So the initial energy of an object
thrown up from, say the surface of the earth would be KE + PE, which must
equal the final energy:
(KE + PE)initial = (KE + PE)final
Since the final destination of the object will be infinite, the final potential
energy will be 0 (since PE = GMm/ and 1/ = 0), and the final kinetic
energy will also be zero, since by definition, the escape velocity is the
minimum velocity needed to escape, so once it’s reaches infinite, it no
longer has any velocity. This means that the final kinetic energy will also
be zero, since ‘v’ is zero. So, we’re left with:
(1/2mv2 – GMm/r) = (0 + 0)
Rearranging:
1/2mv2 = GMm/r
Solve for v:
______
v = (2GM/r) (i.e., the square root of 2GM/r)
So, this shows that for an object of any mass traveling a distance r away
from an object, it will not be captured into orbit by a planet with mass M as
long as it travels fast enough.
You can include a supplement about escape velocity into this lesson very
easily by adding a third variable to the cases in Step 3. above:
3c.) Variable speed: Have students discover their “escape” speed from the
system. Use the intermediate, 4 foot distance = double lengths of yarn.
Have them move along the line at varying speeds (walk, fast walk, slow jog,
faster jog, etc.) until they find out how fast they have to move before the
string breaks. They should find that at a slow speed, they will be captured
into orbit, but when they move more quickly, they can “escape” the
gravitational pull of the “sun.” Even more interesting is if at some speed,
they break the string, but before they do, their straight line course is altered
from its original path (i.e., their trajectory follows an open curve – parabola
or hyperbole – so they feel the force of gravity which alters their straightline path – Newton’s 1st Law, but given their velocity, mass, and distance
from the “sun”, they are not fully captured into orbit).
References
Lesson by Kelly Denney and Katie Schlesinger from The Ohio State
University