Circular and Satellite Motion Legacy Problem #25 Guided Solution
Problem*
Scientists determine the masses of planets by observing the effect of the gravitational field of those planets on nearby objects - mainly upon their moons. By measuring the orbital period and orbital radius of a moon about a planet, Newton's laws of motion can be used to determine the mass of the planet. Phobos, a moon of the planet Mars, was discovered in 1877. Its orbital radius is 9380 km and its orbital period is 0.319 days (2.77 x 104 seconds). Determine the mass of Mars based on this data.
Audio Guided Solution
Have you ever wondered how scientists have determined the mass of the planets or the sun? Well, one of the outcomes of Newton's Law of Universal Gravitation is that scientists can take measurable quantities such as the radius of orbit and the period of orbit and use these measurable quantities in order to determine the mass of the planets or of the sun or of any celestial body about which there is a satellite. For instance, we know the mass of Mars by observing the orbital radius and orbital period of the satellites of Mars. For instance, one of the satellites, Phobos, has an orbital period of 0.319 days in seconds, that would be 2.77 times 10 to the 4th seconds, and it has an orbital radius of 9,380 kilometers in meters, that would be 9.38 times 10 to the 6th meters. Now according to Newton's Law of Universal Gravitation, the speed of an orbiting satellite is simply the square root of big G times the mass of the object being orbited divided by the radius of orbit. The speed of an orbiting satellite, if we presume circular orbits, is also 2 pi r divided by t. Now we can equate these two expressions for the speed of an orbiting satellite and then do some algebra and we would end up with an equation that is stated as t squared over r cubed is equal to 4 pi squared divided by G times the mass of the planet. So there you have an equation that has three variables in it, the period, the radius, and the mass of the object being orbited, the mass of the planet here in this case. So what we can do is rearrange that equation in order to get the mass of Mars by itself, and that becomes the mass of Mars equals 4 times pi squared times the radius of orbit cubed divided by G, big G, and divided by period squared. Substituting in the period in seconds and the radius in meters will give us the mass of Mars. It comes out to be about 6.36 times 10 to the 23rd kilograms.
Solution
6.36 x 1023 kg
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities in an organized manner. Equate given values to the symbols used to represent the corresponding quantity - e.g., \(\descriptive{m}{m,mass} = 61.7\unit{kg}\), \(\descriptive{v}{v,velocity} = 18.5 \unit{\meter\per\second}\), \(\descriptive{R}{R,radius} = 30.9\unit{m}\), \(F_\text{norm} = \colorbox{gray}{Unknown}\).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
Read About It!
Get more information on the topic of Circular and Satellite Motion at The Physics Classroom Tutorial.