Chapter 13: Q. 24 (page 354)

The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of $5.0$ earth years. What are the asteroid’s orbital radius and speed?

Short Answer

Expert verified

The asteroid’s orbital radius is $4.365 \times 10^{11} m$ and its speed is $1.74 \times 10^{4} m / s .$

Step by step solution

Given information

The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of $5$ earth years.

Calculation

The radius can be calculated using the time period and it is given by :

T = \frac{2 π R}{v} R = \frac{T v}{2 π} R = \frac{(1.5768 s \times 10^{8} s) v}{2 π} R = (2.5 \times 10^{7}) v______(1)

Sun's gravitational attraction provides the asteroid's centripetal force and it is given by :

$\frac{G M_{s} M_{a}}{R^{2}} = \frac{M_{a} \times v^{2}}{R} v^{2} = \frac{G \times M_{s}}{R}______(2)$

Continuation of calculation

Substituting the value of R in equation (2) we get,

$v^{2} = \frac{G \times M_{s}}{2.5 \times 10^{7} v} v^{3} = \frac{G \times M_{s}}{2.5 \times 10^{7}} v^{3} = \frac{(6.67 \times 10^{- 11} N m^{2} / k g^{2}) (1.989 \times 10^{30} k g)}{2.5 \times 10^{7}} v^{3} = 5.287 \times 10^{12} m^{3} / s^{3} v = 1.74 \times 10^{4} m / s$

Substituting the value of 'v' in equation (1) we get,

$R = (2.5 \times 10^{7} s) (1.74 \times 10^{4} m / s) R = 4.365 \times 10^{11} m$

Final answer

The asteroid’s orbital radius is $1.74 \times 10^{4} m / s$ and its speed is $4.365 \times 10^{11} m .$

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The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of $5.0$ earth years. What are the asteroid’s orbital radius and speed?

Short Answer

Step by step solution

Given information

Calculation

Continuation of calculation

Final answer

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The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 5.0earth years. What are the asteroid’s orbital radius and speed?

Short Answer

Step by step solution

Given information

Calculation

Continuation of calculation

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Radiation

Engineering Physics

Waves Physics

Magnetism and Electromagnetic Induction

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of $5.0$ earth years. What are the asteroid’s orbital radius and speed?