Chapter 21: Problem 52
A star in a circular orbit about the black hole at the center of the Milky Way (whose mass \(M_{\mathrm{BH}}=8 \times 10^{36} \mathrm{kg}\) ) has an orbital radius of 0.0131 light-year \(\left(1.24 \times 10^{14}\) meters) \right. What is the average speed of this star in its orbit?
Short Answer
Expert verified
The star's average speed is approximately \( 2.075 \times 10^6 \mathrm{m/s} \).
Step by step solution
01
- Understand the given data
The problem states that the mass of the black hole at the center of the Milky Way is given by \(M_{\mathrm{BH}} = 8 \times 10^{36} \mathrm{kg}\). The orbital radius of the star is \0.0131\ light-year, which is equivalent to \(1.24 \times 10^{14}\ meters\).
02
- Use Newton's Law of Universal Gravitation
According to Newton's Law of Universal Gravitation, the gravitational force providing the centripetal force is given by: \[ F = \frac{GM_{\mathrm{BH}}m}{r^2} \] where \(G \) is the gravitational constant \( ( 6.67430 \times 10^{-11} \, \mathrm{m^3} \mathrm{kg}^{-1} \mathrm{s}^{-2} )\), \( m \) is mass of the star, and \( r \) is the orbital radius.
03
- Equate gravitational force and centripetal force
The centripetal force required to keep the star in orbit is given by: \[ F = \frac{mv^2}{r} \]. Equate this to the gravitational force: \[ \frac{GM_{\mathrm{BH}}m}{r^2} = \frac{mv^2}{r} \]
04
- Solve for orbital speed
Cancel out \( m \) on both sides and solve for the orbital speed \( v \): \[ v^2 = \frac{GM_{\mathrm{BH}}}{r} \] \[ v = \sqrt{\frac{GM_{\mathrm{BH}}}{r}} \]
05
- Substitute the known values and calculate
Substitute \( G = 6.67430 \times 10^{-11} \, \mathrm{m^3} \, \mathrm{kg}^{-1} \, \mathrm{s}^{-2} \), \( M_{\mathrm{BH}} = 8 \times 10^{36} \, \mathrm{kg} \), and \( r = 1.24 \times 10^{14} \, \mathrm{m} \) into the equation: \[ v = \sqrt{\frac{(6.67430 \times 10^{-11})(8 \times 10^{36})}{1.24 \times 10^{14}}} \] \[ v = \sqrt{4.305152 \times 10^{12}} \] \[ v \approx 2.075 \times 10^6 \, \mathrm{m/s} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gravitational Force
Gravitational force is a fundamental force of nature that attracts two masses towards each other. The strength of this force depends on the masses of the objects and the distance between them. According to Newton's Law of Universal Gravitation, the force can be calculated using the formula: \[ F = \frac{GM_{1}M_{2}}{r^2} \] where:
- \( F \) is the gravitational force.
- \( G \) is the gravitational constant \((6.67430 \times 10^{-11} \mathrm{m^3 kg^{-1} s^{-2}})\).
- \( M_{1} \) and \( M_{2} \) are the masses of the objects.
- \( r \) is the distance between the centers of the two masses.
Centripetal Force
Centripetal force is the force that keeps an object moving in a circular path. It acts towards the center of the circle. When an object like a star orbits around a black hole, centripetal force is what keeps it on its curved path. It is given by the formula: \[ F = \frac{mv^2}{r} \]where:
- \( F \) is the centripetal force.
- \( m \) is the mass of the object.
- \( v \) is the orbital speed of the object.
- \( r \) is the radius of the circular path (orbital radius).
Newton's Law of Universal Gravitation
Newton's Law of Universal Gravitation states that every mass exerts an attractive force on every other mass. The force between two masses is directly proportional to the product of the masses and inversely proportional to the square of the distance between their centers. To put it mathematically: \[ F = \frac{GM_{1}M_{2}}{r^2} \]where:
- \( M_{1} \) and \( M_{2} \) are the masses of the two objects.
- \( r \) is the distance between the centers of the two masses.
- \( G \) is the gravitational constant \((6.67430 \times 10^{-11} \mathrm{m^3 kg^{-1} s^{-2}})\).
Black Hole
A black hole is a region in space where the gravitational pull is so intense that nothing, not even light, can escape from it. Black holes are formed from the remnants of massive stars that collapse under their own gravity. They have fascinating properties:
- Event Horizon: The boundary around a black hole beyond which no information or matter can escape.
- Singularity: The core of a black hole where the mass is concentrated, leading to infinite density and gravitational pull.
- Gravitational Effects: Black holes can significantly warp space-time and have strong gravitational effects on nearby objects.