Question

Pluto's diameter is approximately 2370 km, and the diameter of its satellite Charon is 1250 km. Although the distance varies, they are often about 19,700 km apart, center to center. Assuming that both Pluto and Charon have the same composition and hence the same average density, find the location of the center of mass of this system relative to the center of Pluto.

Short answer

The center of mass is approximately 2,509 km from the center of Pluto.

Step-by-step solution

Understanding the Center of Mass Formula

The center of mass of two objects along a straight line can be found using the formula \( R = \frac{m_1x_1 + m_2x_2}{m_1 + m_2} \). Here, \( m_1 \) and \( m_2 \) are the masses of Pluto and Charon, respectively, and \( x_1 \) and \( x_2 \) are their positions. Since we need the center of mass relative to Pluto, we can set Pluto's position \( x_1 = 0 \), and Charon's position \( x_2 = d \) where \( d = 19,700 \) km.

Express Mass in Terms of Volume and Density

Assuming identical composition, Pluto and Charon have the same density \( \rho \). The mass of a sphere is given by \( m = \rho \times V = \rho \times \frac{4}{3}\pi r^3 \), where \( r \) is the radius. Let \( r_1 = 1185 \) km (Pluto's radius) and \( r_2 = 625 \) km (Charon's radius). Thus, \( m_1 = \rho \times \frac{4}{3}\pi (1185)^3 \) and \( m_2 = \rho \times \frac{4}{3}\pi (625)^3 \).

Calculate the Mass Ratio

Since \( \rho \) and \( \frac{4}{3}\pi \) are common factors, the mass ratio \( \frac{m_2}{m_1} = \left(\frac{625}{1185}\right)^3 \). Substituting the values, \( \frac{m_2}{m_1} = \left(\frac{625}{1185}\right)^3 = \left(0.527 \right)^3 \approx 0.146 \).

Calculate the Location of the Center of Mass

We use the formula for the center of mass with \( x_1 = 0 \) and \( x_2 = 19,700 \). The center of mass \( R \) relative to Pluto is \( R = \frac{0 \cdot m_1 + 19,700 \cdot m_2}{m_1 + m_2} = \frac{19,700 \cdot m_2}{m_1 + m_2} \). Substitute the mass ratio \( \frac{m_2}{m_1} = 0.146 \), thus \( m_2 = 0.146m_1 \). Therefore, \[ R = \frac{19,700 \times 0.146m_1}{m_1(1 + 0.146)} = \frac{19,700 \times 0.146}{1.146} \].

Solve for R

Substitute and solve \( \frac{19,700 \times 0.146}{1.146} \approx 2,509 \). Thus, the center of mass location is approximately 2,509 km from the center of Pluto.

Key concepts

Pluto and Charon

Pluto and Charon are one of the most interesting pairs in our solar system due to their unique characteristics. Pluto, once considered the ninth planet, is now classified as a dwarf planet. It orbits the Sun from a great distance and is primarily composed of ice and rock. Its notable satellite, Charon, is quite substantial relative to Pluto's size. Charon's diameter is about half that of Pluto's, making the system more like a double dwarf planet system, rather than a typical planet-moon pair.
Pluto's diameter is around 2370 km, while Charon measures about 1250 km. They orbit a common center of gravity located between them, unlike the Earth-Moon system, where the center of mass is within Earth itself. This unique feature makes their gravitational dance quite intriguing. The distance from the center of Pluto to the center of Charon is approximately 19,700 km. This orbital relationship is key in calculating many of their orbital properties, including their center of mass.

Mass Ratio

Understanding the mass ratio of Pluto and Charon is crucial in many physics problems. The mass of an object is dependent on its density (assumed constant for this problem) and its volume. Since both Pluto and Charon are assumed to have the same composition, they have the same average density, which allows us to calculate their mass ratio using their volumes alone.
The mass ratio is derived from their volumes, with volume proportional to the cube of the radius. Thus, the formula for the mass ratio becomes \[ \left(\frac{625}{1185}\right)^3 \approx 0.146 \].
This tells us that Charon's mass is about 14.6% that of Pluto's, a surprisingly large fraction given its smaller size. This mass ratio is important for calculating the center of mass of the two bodies in their shared orbital system.

Density and Volume

Density and volume are fundamental concepts in this physics problem. Density, denoted as \( \rho \), is mass per unit volume, a measure of how compact the mass in an object is. Volume, on the other hand, represents the space that an object occupies. For spheres like Pluto and Charon, volume can be calculated with the formula:\[ V = \frac{4}{3}\pi r^3 \]
where \( r \) is the radius.
Assuming Pluto and Charon have the same density due to identical composition, their masses can be expressed with \[ m = \rho \times V \].
Hence, their masses depend directly on the cube of their radii. This relationship allows us to simplify the calculation of the forces, ratios, and center of mass without needing to know the actual density. It elegantly reduces the complex problem into manageable calculations involving ratios of their radii.

Physics Problem Solving

When tackling problems like the location of the center of mass of the Pluto-Charon system, a systematic approach simplifies complex concepts. Starting with understanding the problem setup is crucial, such as knowing what the system configuration is (i.e., Pluto and Charon modeled as two point masses on a line).
Next, apply relevant formulas like the center of mass formula \[ R = \frac{m_1x_1 + m_2x_2}{m_1 + m_2} \], selecting suitable references such as setting one position as zero. For this problem, Pluto's center was set as the origin. Find expressions for unknowns, such as masses based on volume. Calculate relevant ratios, like \( \frac{m_2}{m_1} \), which simplifies the use of the mass in the center of mass equation. Once arranged, substitute known values and solve the equation step-by-step to find the desired quantity, such as the center of mass location. This methodical, formula-driven approach aids in solving physics problems efficiently and accurately.