Students construct a model of an object in orbit by balancing centripetal force, or circular force, with gravitational force.
- Consider limiting group sizes to 3-4 to keep all students actively engaged.
- Using lighter objects for the hanging and swinging masses means that students will not have to spin their model as quickly to achieve the desired results. Objects such as erasers, golf balls, and tennis balls are the size, hardness, and weight that should be used.
- For middle school students, the math may be too complex, but the balancing of forces can be a valuable model.
Despite the incredible complexity of building and launching satellites, the principles that govern how to keep an object in orbit have been known in great detail for hundreds of years. Isaac Newton tested his Laws of Motion, published in 1687, with simple objects here on Earth. But even then, he proposed (and scientists later verified) that the laws extend well "into the heavens," describing the motions of planets, our Moon, and as a result, human-made satellites that orbit Earth and other planets.
To get a satellite into orbit above Earth, we first need to launch that satellite on a rocket. Once in orbit, we want the satellite to be traveling not so fast that it escapes Earth’s gravity entirely, but also not so slow that the gravity of Earth pulls it back downward. Engineers have to make careful calculations when designing trajectories to find a balance between the gravitational pull of Earth and the circular force of the satellite. When perfectly balanced, these two competing forces allow satellites to stay in orbit as they simultaneously fall downward toward Earth and fly past Earth at exactly the same rate.
NASA engineers – and students – can use Newton's laws to predict the gravitational and centripetal forces at any given distance from Earth or speed of a satellite. This is what allows NASA engineers to design orbits for the numerous spacecraft orbiting Earth and other planets today.
- Describe to students how the balance or imbalance of forces acting on an object determines how the object will move. Note that even objects at rest have forces acting on them – the forces just add up to zero. Similarly, an object orbiting Earth is moving, but not speeding up or slowing down, nor crashing into Earth or being flung out into the solar system. The equal and opposite nature of these balanced forces, described by Newton’s third law, gives us a glimpse into how we can use math to put satellites into orbit or explain the motion of objects in our solar system.
- Divide students into groups and distribute the student handouts and a set of materials to each group. Tell students they will be creating a model of an object in orbit to calculate the forces acting on the object.
- Following the instructions on their worksheet, have students record the weight of the object they will be using as their known hanging mass.
- Have students run the string through the straw, and tie one end of the string around the object that will become the hanging mass. Be sure that the string is tied securely enough that it can withstand some movement.
- Students should then attach an object of unknown swinging mass to the other end of the string, securing tightly.
- Holding the straw vertically, with the object of unknown swinging mass on top, students should begin to gently move their devices in a circle so that the unknown mass spins around the straw like helicopter blades.
- Have them continue this movement, experimenting with different speeds to get the unknown mass spinning fast enough that the known hanging mass is neither rising nor falling. Remind students not to pinch the straw, restricting the movement of the string.
- Once students have a feel for how quickly they need to spin their devices, have a group member time several complete revolutions. Note that it may be easier to time 10 or 20 cycles and divide to get the average time for a single cycle. Have students record the time on the worksheet.
- Once the trial is complete, have the student spinning the device pinch the straw closed while the masses are at equilibrium. Have them record the distance from the top of their straw to the unknown swinging mass as the radius of the orbit.
- On their worksheets, student groups should now be able to calculate:
- The gravitational force from the known hanging mass.
- The centripetal force using the time and radius they recorded, given that Fg = Fc (see worksheet).
- The mass of the unknown object.
- Have students weigh the unknown mass and compare their prediction with the true value.
- What critical scenario is taking place when the hanging object is not falling or rising?
- How did the predicted mass of the unknown object compare to its true value? What could have given rise to the differences observed?
- In true planetary orbits, we’re looking not at the force of gravity pulling an object toward the ground, but the force of gravity between the two objects. Recall that gravitational force is given by the equation:
Fg = GM1M2 / R2
- Knowing that centripetal force and gravitational force need to be equal for an orbit to take place, what variables cancel each other out? What parts of the simplified equation remain?
M1V2 = GM1M2 / R
V = sqrt(M2/R)
- Therefore, knowing the mass of the object orbiting is not actually important when calculating an orbit!
- Students should be able to accurately calculate centripetal acceleration and thus rotational velocity from the force of gravity of the hanging object.
- Calculations should be captured in the student worksheet, showing that the gravitational force is balanced against the centripetal force.
For an additional challenge, consider having students connect their force calculations to kinetic and potential energy. Both can be calculated using the velocity and height of their model. What errors are introduced here that could explain why these numbers are not equivalent?
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