We are given two formulas that are to be set equal to each other: the universal gravitational formula and the formula for centripetal force.

1. Universal Gravitational Formula: $F = \frac{G M m}{r^2}$ where:

   • $F$ is the gravitational force,
   • $G$ is the gravitational constant,
   • $M$ is the mass of the larger object (e.g., a planet),
   • $m$ is the mass of the smaller object (e.g., a satellite),
   • $r$ is the distance between the two objects.

2. Centripetal Force Formula: $F = \frac{m v^2}{r}$ where:

   • $F$ is the centripetal force,
   • $m$ is the mass of the smaller object,
   • $v$ is the velocity of the object,
   • $r$ is the radius (same as in the gravitational formula).

Setting these two forces equal to each other:

$\frac{G M m}{r^2} = \frac{m v^2}{r}$

Step 1: Simplify the equation

We can cancel out $m$ from both sides of the equation, assuming $m \neq 0$:

$\frac{G M}{r^2} = \frac{v^2}{r}$

Step 2: Solve for $v$

Multiply both sides of the equation by $r$ to eliminate the denominator on the right-hand side:

$\frac{G M}{r} = v^2$

Step 3: Take the square root of both sides:

$v = \sqrt{\frac{G M}{r}}$

Final Answer:

$v = \sqrt{\frac{G M}{r}}$

This is the velocity of an object in orbit under the influence of gravity.

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Would you like a deeper explanation of any part of this? Here are some related questions you could explore:

1. How does this velocity change if the distance $r$ increases?
2. What happens to the velocity if the mass $M$ of the planet doubles?
3. How does this formula relate to the concept of escape velocity?
4. Can this formula be used to calculate the speed of planets around the sun?
5. How would you calculate the orbital period from this velocity?

Tip: The gravitational constant $G$ is very small, so it often requires large masses (like planets) to produce significant gravitational forces.