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Six artificial satellites circle a space station at constant speed. The mass m of each satellite,...

Six artificial satellites circle a space station at constant speed. The mass m of each satellite, distance L from the space station, and the speed v of each satellite are listed below. The satellites fire rockets that provide the force needed to maintain a circular orbit around the space station. The gravitational force is negligible. 1. m=200kg,L= 5000 m, v=120 m/s 2. m=800kg,l= 10,000 m, v=40m/s 3. m=400kg,L= 2500 m, v=80m/s 4. m=100kg, L=2500 m, V=160m/s 5. m=300kg, L=10,000m, V=80 m/s 6. m=200kg, L=5000 m, V=160 m/s Part A Rank each satellite from largest to smallest based on its period. Part B Rank each satellite from largest to smallest based on its acceleration.
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Answer #1
Concepts and reason

The concepts used to solve the problem are centripetal acceleration and period of motion of an object moving in the circular path.

Initially, calculate the time period for all the satellites by using the expression of period of motion and then rank them from largest to smallest from largest to smallest based on its period.

Later, calculate the acceleration for all the satellites by using the expression for the acceleration and rank them form largest to smallest.

Fundamentals

Consider an object of mass m moving in circular path of radius L. The period of revolution T of the object is given as follows:

T=2πLvT = \frac{{2\pi L}}{v} …… (1)

Here, v is the constant speed of the object.

The acceleration a of the object is given as follows:

a=v2La = \frac{{{v^2}}}{L}

(a)

Calculate the period of satellite having mass 200 kg.

For the period of revolution of first satellite T1{T_1} , substitute 5000m5000{\rm{ m}} for L and 120ms1120{\rm{ m}} \cdot {{\rm{s}}^{ - 1}} for v in equation (1).

T1=2π(5000m)120ms1=2(3.141)(5000)160=261.67s\begin{array}{c}\\{T_1} = \frac{{2\pi \left( {5000{\rm{ m}}} \right)}}{{120{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}}}\\\\ = \frac{{2\left( {3.141} \right)\left( {5000} \right)}}{{160}}\\\\ = 261.67{\rm{ s}}\\\end{array}

Calculate the period of satellite having mass 800 kg.

Substitute 10,000m10,000{\rm{ m}} for L and 40ms140{\rm{ m}} \cdot {{\rm{s}}^{ - 1}} for v in equation (1).

T2=2π(10,000m)40ms1=1570.0s\begin{array}{c}\\{T_2} = \frac{{2\pi \left( {10,000{\rm{ m}}} \right)}}{{40{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}}}\\\\ = 1570.0{\rm{ s}}\\\end{array}

Calculate the period of satellite having mass 400 kg.

Substitute 2500m2500{\rm{ m}} for L and 80ms180{\rm{ m}} \cdot {{\rm{s}}^{ - 1}} for v in equation (1).

T3=2π(2500m)80ms1=196.25s\begin{array}{c}\\{T_3} = \frac{{2\pi \left( {2500{\rm{ m}}} \right)}}{{80{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}}}\\\\ = 196.25{\rm{ s}}\\\end{array}

Calculate the period of satellite having mass 100 kg.

Substitute 2500m2500{\rm{ m}} for L and 160ms1160{\rm{ m}} \cdot {{\rm{s}}^{ - 1}} for v in equation (1).

T4=2π(2500m)160ms1=98.125s\begin{array}{c}\\{T_4} = \frac{{2\pi \left( {2500{\rm{ m}}} \right)}}{{160{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}}}\\\\ = 98.125{\rm{ s}}\\\end{array}

Calculate the period of satellite having mass 300 kg.

Substitute 10,000m10,000{\rm{ m}} for L and 80ms180{\rm{ m}} \cdot {{\rm{s}}^{ - 1}} for v in equation (1).

T5=2π(10000m)80ms1=785.0s\begin{array}{c}\\{T_5} = \frac{{2\pi \left( {10000{\rm{ m}}} \right)}}{{80{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}}}\\\\ = 785.0{\rm{ s}}\\\end{array}

Calculate the period of satellite having mass 200 kg.

Substitute 5,000m5,000{\rm{ m}} for L and 160ms1160{\rm{ m}} \cdot {{\rm{s}}^{ - 1}} for v in equation (1).

T6=2π(5,000m)160ms1=196.25s\begin{array}{c}\\{T_6} = \frac{{2\pi \left( {5,000{\rm{ m}}} \right)}}{{160{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}}}\\\\ = 196.25{\rm{ s}}\\\end{array}

(b)

Find the acceleration of the satellites.

The acceleration of the satellite is given by following expression:

a=v2La = \frac{{{v^2}}}{L} …… (2)

Calculate the acceleration of satellite having mass 200 kg.

Substitute 5000m5000{\rm{ m}} for L and 120ms1120{\rm{ m}} \cdot {{\rm{s}}^{ - 1}} for v in equation (2).

a1=(120ms1)25000m=2.88ms2\begin{array}{c}\\{a_1} = \frac{{{{\left( {120{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}} \right)}^2}}}{{5000{\rm{ m}}}}\\\\ = 2.88{\rm{ m}} \cdot {{\rm{s}}^{ - 2}}\\\end{array}

Calculate the acceleration of satellite having mass 800 kg.

Substitute 10,000m10,000{\rm{ m}} for L and 40ms140{\rm{ m}} \cdot {{\rm{s}}^{ - 1}} for v in equation (2).

a2=(40ms1)210,000m=0.16ms2\begin{array}{c}\\{a_2} = \frac{{{{\left( {40{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}} \right)}^2}}}{{10,000{\rm{ m}}}}\\\\ = 0.16{\rm{ m}} \cdot {{\rm{s}}^{ - 2}}\\\end{array}

Calculate the acceleration of satellite having mass 400 kg.

Substitute 2500m2500{\rm{ m}} for L and 80ms180{\rm{ m}} \cdot {{\rm{s}}^{ - 1}} for v in equation (2).

a3=(80ms1)22500m=2.56ms2\begin{array}{c}\\{a_3} = \frac{{{{\left( {{\rm{80 m}} \cdot {{\rm{s}}^{ - 1}}} \right)}^2}}}{{2500{\rm{ m}}}}\\\\ = 2.56{\rm{ m}} \cdot {{\rm{s}}^{ - 2}}\\\end{array}

Calculate the acceleration of satellite having mass 100 kg.

Substitute 2500m2500{\rm{ m}} for L and 160ms1160{\rm{ m}} \cdot {{\rm{s}}^{ - 1}} for v in equation (2).

a4=(160ms1)22500m=10.24ms2\begin{array}{c}\\{a_4} = \frac{{{{\left( {160{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}} \right)}^2}}}{{2500{\rm{ m}}}}\\\\ = 10.24{\rm{ m}} \cdot {{\rm{s}}^2}\\\end{array}

Calculate the acceleration of satellite having mass 300 kg.

For the acceleration a5{a_5} of fifth satellite, substitute 10,000m10,000{\rm{ m}} for L and 80ms180{\rm{ m}} \cdot {{\rm{s}}^{ - 1}} for v in equation (2).

a5=(80ms1)210,000m=0.64ms2\begin{array}{c}\\{a_5} = \frac{{{{\left( {80{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}} \right)}^2}}}{{10,000{\rm{ m}}}}\\\\ = 0.64{\rm{ m}} \cdot {{\rm{s}}^{ - 2}}\\\end{array}

Calculate the acceleration of satellite having mass 200 kg.

The acceleration a6{a_6} of sixth satellite is calculated by substituting 5,000m5,000{\rm{ m}} for L and 160ms1160{\rm{ m}} \cdot {{\rm{s}}^{ - 1}} for v in equation (2).

a6=(160ms1)25000m=5.12ms2\begin{array}{c}\\{a_6} = \frac{{{{\left( {160{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}} \right)}^2}}}{{5000{\rm{ m}}}}\\\\{\rm{ = 5}}{\rm{.12 m}} \cdot {{\rm{s}}^2}\\\end{array}

Ans: Part a

The order of satellites largest to smallest based on their period is T2>T5>T1>T3=T6>T4{T_2} > {T_5} > {T_1} > {T_3} = {T_6} > {T_4} .

Part b

The order of satellites largest to smallest based on their acceleration is a4>a6>a1>a3>a5>a2{a_4} > {a_6} > {a_1} > {a_3} > {a_5} > {a_2} .

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