A student trying to calculate the parameters of a satellite orbit obtained a quantity Q which is related with the orbital radius
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,
radius of the earth
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and acceleration due to gravity,
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by the formula
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=
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2
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½
. Using dimensional analysis, find out what Q represents.
Select one:
Tangential force
Torque on the satellite
Centripetal acceleration
Orbital speed
Using dimensional analysis, we can determine the dimensions of Q by analyzing the dimensions of the variables involved in the formula.
Let's start with the numerator:
- The square of the orbital radius has dimensions of length squared (L^2).
Now let's move on to the denominator:
- The radius of the earth has dimensions of length (L).
- The acceleration due to gravity has dimensions of length per time squared (LT^-2).
Putting it all together, we have:
Q = (L^2) / (L * LT^-2)^1/2
Simplifying the denominator:
Q = (L^2) / (L^2T^-2)^1/2
Q = (L^2) / (L)T^-1
Q = LT
Therefore, Q represents a quantity with dimensions of length times time, which is the unit for speed.
Therefore, the correct answer is: Orbital speed.
Q represents the Centripetal acceleration, but don't worry, I won't spin you in circles with a long explanation!
To determine what quantity Q represents, we can analyze the dimensions of each term in the formula using dimensional analysis.
Let's break down the dimensions of each quantity:
The orbital radius, R, has units of length (L).
The radius of the Earth, r, also has units of length (L).
The acceleration due to gravity, g, has units of acceleration (L/T^2), where T represents time.
Now let's analyze the dimensions of the formula:
Q = (R * r * g)^(2/3)
Q has unknown units since it is the quantity we are trying to determine.
Next, let's break down the dimensions for each term:
R * r * g = (L) * (L) * (L/T^2) = L^3/T^2
(R * r * g)^(2/3) = (L^3/T^2)^(2/3) = L^2/T^(4/3)
From the dimensional analysis, we can see that Q has dimensions of length squared divided by time to the power of 4/3 (L^2/T^(4/3)).
Out of the four given options, the only option that corresponds to these dimensions is "Orbital speed." Therefore, Q represents the orbital speed of the satellite.
To determine what quantity Q represents using dimensional analysis, we need to analyze the units of the terms involved in the formula.
Let's break down the formula:
Q = (radius of the satellite orbit) / (radius of the earth) √(acceleration due to gravity)
We know that:
- Radius has the unit of length (meters, km, etc.).
- Acceleration due to gravity has the unit of acceleration (m/s^2).
Let's analyze each term separately:
- The radius of the satellite orbit has the unit of length.
- The radius of the earth also has the unit of length.
- The acceleration due to gravity has the unit of acceleration.
To simplify the equation, we need to cancel out similar units in the numerator and denominator.
Looking at the terms, we see that the radius of the earth in the denominator cancels out with the radius of the satellite orbit in the numerator, leaving us with just a unit of length:
Q = (radius of the satellite orbit) * (√(acceleration due to gravity))
=> Q = [length] * [length]
=> Q = [length]^2
Now, based on the units, we can conclude that Q represents an area, which is expressed in square units (such as m^2, km^2, etc.).
Therefore, the correct choice is that Q represents the orbital area, or in other words, it represents the area enclosed by the satellite orbit around the earth.