PHET ONLINE LAB MY SOLAR SYSTEM WORKSHEET PLEASE HELP Activity 68 ?Select the “Sun and Planet” preset from the drop-down menu in the upper right. Cllckstart and observe the motion of Body 1, ?in this system, the mass of the small body is not insignificant relative to the larger body. Both bodies orbit their center of mass. In this activity you will create a new 2 -body system where each body orbits in a circle about the center of mass as shown in the figure at right. The radius of each orblt is equal to the distance between the center of the body and the center of mass. Body 1 ?has a mass of 300????×1028(??). ?Body 2 ?has a mass of 75????×1028 (??2). ?Their centers are separated by 4 ?AU. ?Calculate the distance ??1 ?is from the center of mass.
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This distance will be the radius of the smaller circular orbit. What is the radius of ??2 ‘s larger orbit? Show all your work below:
The correct answer and explanation is:
To calculate the distances from the center of mass (COM) to the two bodies, we use the center of mass formula for a two-body system: r1=m2m1+m2×d,r2=m1m1+m2×dr_1 = \frac{m_2}{m_1 + m_2} \times d, \quad r_2 = \frac{m_1}{m_1 + m_2} \times d
Where:
- r1r_1 = distance of Body 1 from the center of mass
- r2r_2 = distance of Body 2 from the center of mass
- m1m_1 = mass of Body 1
- m2m_2 = mass of Body 2
- dd = distance between the two bodies (4 AU)
Given Data:
- m1=300×1028 kgm_1 = 300 \times 10^{28} \, \text{kg}
- m2=75×1028 kgm_2 = 75 \times 10^{28} \, \text{kg}
- d=4 AUd = 4 \, \text{AU}
Step 1: Calculate r1r_1
r1=m2m1+m2×dr_1 = \frac{m_2}{m_1 + m_2} \times d
Substitute the values: r1=75×1028300×1028+75×1028×4r_1 = \frac{75 \times 10^{28}}{300 \times 10^{28} + 75 \times 10^{28}} \times 4
Simplify: r1=75375×4r_1 = \frac{75}{375} \times 4 r1=15×4=0.8 AUr_1 = \frac{1}{5} \times 4 = 0.8 \, \text{AU}
Step 2: Calculate r2r_2
r2=m1m1+m2×dr_2 = \frac{m_1}{m_1 + m_2} \times d
Substitute the values: r2=300×1028300×1028+75×1028×4r_2 = \frac{300 \times 10^{28}}{300 \times 10^{28} + 75 \times 10^{28}} \times 4
Simplify: r2=300375×4r_2 = \frac{300}{375} \times 4 r2=45×4=3.2 AUr_2 = \frac{4}{5} \times 4 = 3.2 \, \text{AU}
Final Answers:
- r1=0.8 AUr_1 = 0.8 \, \text{AU} (smaller orbit radius)
- r2=3.2 AUr_2 = 3.2 \, \text{AU} (larger orbit radius)
Explanation:
The center of mass (COM) is the point where the combined mass of the system balances. It is closer to the more massive body, m1m_1, due to its larger mass. The calculation uses the ratio of the masses to find how far each body is from the COM. Here, Body 1’s orbit radius (r1r_1) is smaller because it is significantly heavier than Body 2. Body 2, being lighter, orbits farther away from the COM. These circular orbits represent the gravitational balance between the two bodies as they revolve around their mutual center of mass.