Question

Rank each satellite based on the net force acting on it. Rank from largest to smallest.

1. m= 100kg, L = 2500m, v= 160 m/s
2. m= 300kg L = 10000m, v= 80 m/s
3. m = 400kg L = 2500m, v= 80 m/s
4. m = 200kg L = 5000m, v= 160 m/s
5. m = 800kg L = 10000m, v= 40 m/s
6. m = 200kg L = 5000m, v= 120 m/s

Answer 1

Concepts and reason

The required concept to solve the problem is the centripetal force.

First, find the net force acting on each of the satellite. Then, rank the satellite from largest to smallest based on the net force acting on it.

Fundamentals

A satellite’s motion is a projectile motion. The only force acting on the satellite is gravity. As the satellite is moving in a circular motion, the force acting on the satellite is the centripetal force. This centripetal force is provided by gravity. Centripetal force is given by

${F_C} = \frac{{m{v^2}}}{r}$

Here, $m$ is the mass of the satellite, $v$ is the velocity of the satellite, and$r$ is the radius of the circular orbit.

As the motion of the satellite is circular, the velocity of the particle will be directed tangent to the circle at every point along the path. The acceleration is directed towards the center of the circle.

The only force acting on a satellite is the centripetal force provided by gravity. So,

$\begin{array}{c}\\{F_{net}} = {F_C}\\\\ = \frac{{m{v^2}}}{L}\\\end{array}$

Here, $m$is the mass of the satellite, $v$is the velocity of the satellite and$L$is the radius of the circular orbit.

For the first satellite:

${F_1} = \frac{{{m_1}v_1^2}}{{{L_1}}}$

Substitute$100\,{\rm{kg}}$for${m_1}$, $160\,{\rm{m/s}}$for${v_1}$and$2500\,{\rm{m}}$for${L_1}$in the above expression.

$\begin{array}{c}\\{F_1} = \frac{{\left( {100\,{\rm{kg}}} \right){{\left( {160\,{\rm{m/s}}} \right)}^2}}}{{2500\,{\rm{m}}}}\\\\ = 1024\,{\rm{N}}\\\end{array}$

For the second satellite:

${F_2} = \frac{{{m_2}v_2^2}}{{{L_2}}}$

Substitute$300\,{\rm{kg}}$for${m_2}$, $80\,{\rm{m/s}}$for${v_2}$and$10000\,{\rm{m}}$for${L_2}$ in the above expression.

$\begin{array}{c}\\{F_2} = \frac{{\left( {300\,{\rm{kg}}} \right){{\left( {80\,{\rm{m/s}}} \right)}^2}}}{{10000\,{\rm{m}}}}\\\\ = 192\,{\rm{N}}\\\end{array}$

For the third satellite:

${F_3} = \frac{{{m_3}v_3^2}}{{{L_3}}}$

Substitute$400\,{\rm{kg}}$for${m_3}$, $80\,{\rm{m/s}}$for${v_3}$and$2500\,{\rm{m}}$for${L_3}$ in the above expression.

$\begin{array}{c}\\{F_3} = \frac{{\left( {400\,{\rm{kg}}} \right){{\left( {80\,{\rm{m/s}}} \right)}^2}}}{{2500\,{\rm{m}}}}\\\\ = 1024\,{\rm{N}}\\\end{array}$

For the fourth satellite:

${F_4} = \frac{{{m_4}v_4^2}}{{{L_4}}}$

Substitute$200\,{\rm{kg}}$for${m_4}$, $160\,{\rm{m/s}}$for${v_4}$and$5000\,{\rm{m}}$for${L_4}$ in the above expression.

$\begin{array}{c}\\{F_4} = \frac{{\left( {200\,{\rm{kg}}} \right){{\left( {160\,{\rm{m/s}}} \right)}^2}}}{{5000\,{\rm{m}}}}\\\\ = 1024\,{\rm{N}}\\\end{array}$

For the fifth satellite:

${F_5} = \frac{{{m_5}v_5^2}}{{{L_5}}}$

Substitute$800\,{\rm{kg}}$for${m_5}$, $40\,{\rm{m/s}}$for${v_5}$and$10000\,{\rm{m}}$for${L_5}$ in the above expression.

$\begin{array}{c}\\{F_5} = \frac{{\left( {800\,{\rm{kg}}} \right){{\left( {40\,{\rm{m/s}}} \right)}^2}}}{{10000\,{\rm{m}}}}\\\\ = 128\,{\rm{N}}\\\end{array}$

For the sixth satellite:

${F_6} = \frac{{{m_6}v_6^2}}{{{L_6}}}$

Substitute $200\,{\rm{kg}}$ for ${m_6}$, $120\,{\rm{m/s}}$ for${v_6}$ and $5000\,{\rm{m}}$ for ${L_6}$ in the above expression.

$\begin{array}{c}\\{F_6} = \frac{{\left( {200\,{\rm{kg}}} \right){{\left( {120\,{\rm{m/s}}} \right)}^2}}}{{5000\,{\rm{m}}}}\\\\ = 576\,{\rm{N}}\\\end{array}$

From the calculated centripetal forces, it is clear that the satellites 1, 3 and 4 have the same amount of force acting on it, which is the largest force. The satellite 6 has force acting on it which is the second largest. Satellite 2 has a slightly lower value of force than that of satellite 6 and satellite 5 which has the least amount of force acting on it. Hence, the ranking of the satellites from largest to smallest, based on the force acting on them is

$1 = 3 = 4 > 6 > 2 > 5$.

Ans:

The ranking of the satellites from largest to smallest, based on the force acting on them is

$1 = 3 = 4 > 6 > 2 > 5$.