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Chapter 1: Problem 2

The two components of a double star are observed to move in circles of radii \(r_{1}\) and \(r_{2}\). What is the ratio of their masses? (Hint: Write down their accelerations in terms of the angular velocity of rotation, \(\omega .\) )

Short Answer

Expert verified
Question: Calculate the ratio of the masses of the two components of a double star system if they both move in circles of radii \(r_{1}\) and \(r_{2}\). Answer: The ratio of the masses of the two components of the double star system is \(\frac{m_{1}}{m_{2}} = \frac{r_{2}}{r_{1}}\).

Step by step solution

01

Compute the centripetal acceleration of both stars

The centripetal acceleration of an object moving in a circle of radius r with an angular velocity \(\omega\) is given by the formula: `a = rω^2`. For the two components of the double star system, we have: acentripetal acceleration of star 1: \(a_{1} = r_{1}\omega^{2}\) acentripetal acceleration of star 2: \(a_{2} = r_{2}\omega^{2}\)
02

Apply Newton's second law of motion

According to Newton's second law of motion, the force acting on an object is equal to the product of its mass and acceleration. Therefore, we have: force on star 1: \(F_{1} = m_{1}a_{1} = m_{1}r_{1}\omega^{2}\) force on star 2: \(F_{2} = m_{2}a_{2} = m_{2}r_{2}\omega^{2}\)
03

Deduce the ratio of their masses

As the two stars are in a double star system, they exert gravitational forces on each other and hence, \(F_{1} = F_{2}\). Using the expressions we obtained in Step 2, we can write this condition as follows: \(m_{1}r_{1}\omega^{2} = m_{2}r_{2}\omega^{2}\) Now we can find the ratio of their masses by dividing the two equations: \(\frac{m_{1}}{m_{2}} = \frac{r_{2}}{r_{1}}\) This is the final expression for the ratio of the masses of the two components of the double star system in terms of their respective radii.

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