Lesson Plan - Ohio State University
Lesson Plan | |
|Lesson Title |How Gravity Affects Orbits |
|Grade Level | |
|State Indicator(s) | |
|Goals and Objectives for Student |Students will explore how the force of gravity affects celestial bodies to cause one to orbit around the other |
|Learning |(i.e., mimicking the planets orbiting the sun in the solar system). |
| | |
| |Main points to be covered include: |
| |Gravity is always an attractive force operating between any two masses |
| |The strength of the force depends directly on the masses of the two bodies. |
| |The strength of the force depends inversely on the distance between the two objects. |
| |Whether an object is captured into a gravitational orbit depends on the mass, radius, and velocity of the |
| |approaching object. |
|Diversity | |
|Teaching Method |Guided to open inquiry |
|Learning Activities |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: |
| | |
| |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: |
| |[pic] |
| |Provide each student with several 8 ft. lengths (~10) of yarn. |
| |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). |
| |Cases to try: |
| |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. |
| |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 gravitational force on distance. |
| |Have students record their experiences, and discuss and even graph their results. |
| |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. |
|Materials |Skein of yarn (cheap “Red Heart” brand polyester works well) |
| |Masking tape |
| |Yard Stick |
| |Cake donut (for demo.) |
|Supplements |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 straight-line 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 |
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Planet
Sun
8 ft
4 ft
2 ft
START
START
START
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