How do we calculate the "radius of gyration" of a 500g steel rod spinning at 300 RPM and having a lenght of 30cm?
We need this to find the moment of Inertia... don't we?
To calculate the radius of gyration of a steel rod, we first need to understand what it is. The radius of gyration (symbolized as "k") is a measure of how the mass of an object is distributed around its axis of rotation. It represents a hypothetical distance from the axis at which the entire mass of the object can be concentrated to have the same moment of inertia.
To calculate the radius of gyration, we can use the formula:
k = sqrt(I / m)
Where:
- k is the radius of gyration
- I is the moment of inertia
- m is the mass of the object
However, before we can calculate the radius of gyration, we need to find the moment of inertia (I) for the steel rod.
The moment of inertia of a long, thin rod rotating around its center can be found using the formula:
I = (1/12) * m * L^2
Where:
- I is the moment of inertia
- m is the mass of the rod
- L is the length of the rod
Given the following information:
- Mass of the steel rod (m) = 500 grams (or 0.5 kg)
- Length of the steel rod (L) = 30 cm (or 0.3 meters)
First, we calculate the moment of inertia (I) using the above formula:
I = (1/12) * 0.5 kg * (0.3 m)^2
I = (1/12) * 0.5 kg * 0.09 m^2
Next, we find the radius of gyration (k) using the formula mentioned earlier:
k = sqrt(I / m)
k = sqrt((1/12) * 0.5 kg * 0.09 m^2 / 0.5 kg)
k = sqrt((1/12) * 0.09 m^2)
k = sqrt(0.0075 m^2)
k ≈ 0.0866 meters
Therefore, the radius of gyration of the steel rod is approximately 0.0866 meters.