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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 5.0earth years. What are the asteroid鈥檚 orbital radius and speed?

Short Answer

Expert verified

The asteroid鈥檚 orbital radius is4.365脳1011mand its speed is1.74脳104m/s.

Step by step solution

01

Given information

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

02

Calculation

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

T=2蟺RvR=Tv2蟺R=(1.5768s脳108s)v2蟺R=(2.5脳107)v______(1)

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

GMsMaR2=Ma脳v2Rv2=G脳MsR______(2)

03

Continuation of calculation

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

v2=G脳Ms2.5脳107vv3=G脳Ms2.5脳107v3=(6.67脳10-11Nm2/kg2)(1.989脳1030kg)2.5脳107v3=5.287脳1012m3/s3v=1.74脳104m/s

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

R=(2.5脳107s)(1.74脳104m/s)R=4.365脳1011m

04

Final answer

The asteroid鈥檚 orbital radius is 1.74脳104m/sand its speed is4.365脳1011m.

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