A satellite is orbiting around a planet. its velocity is found to depend upon

a.)Radius of orbit
b.)mass of planet
c.)universal gravitational constant(G)
Using dimensional analysis find an expression relating orbital velocity to the above physical quantities

To find an expression relating orbital velocity to the given physical quantities (radius of orbit, mass of planet, and universal gravitational constant), we can use dimensional analysis.

Dimensional analysis is a technique that allows us to derive relationships between physical quantities by considering their units. In this case, we want to find an expression for orbital velocity, so we need to determine how the units of the given quantities relate to the units of velocity.

Let's start by considering the relevant quantities:

- Radius of orbit (r): The unit of radius is length, usually measured in meters (m).
- Mass of planet (M): The unit of mass is mass, usually measured in kilograms (kg).
- Universal gravitational constant (G): The unit of G is the gravitational constant, usually measured in cubic meters per kilogram per second squared (m^3/kg/s^2).

The unit of velocity is length over time, usually measured in meters per second (m/s).

To find an expression relating these quantities, we can consider how the dimensions of the variables combine.

Orbital velocity can be expressed as the square root of the product of gravitational constant (G), mass of the planet (M), divided by the radius of orbit (r), given by the equation:

v = √(G * M / r)

Applying dimensional analysis, we can determine the units for each term in the equation and confirm that they are consistent:

√(G * M / r) = √(m^3/kg/s^2 * kg / m)

Taking the square root, we have:

√(m^3⋅kg⋅kg/(kg⋅m⋅s^2⋅m))

Simplifying the units, we get:

√(m^3⋅kg^2/(kg⋅m⋅s^2))

Cancelling out the kg's and m's:

√(m^2/s^2)

This simplifies to:

m/s

which is the unit of velocity. Therefore, we have derived the expression:

v = √(G * M / r)

where v denotes the orbital velocity, G is the universal gravitational constant, M is the mass of the planet, and r is the radius of the orbit.