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?
Yes, you are correct. In order to calculate the moment of inertia, we first need to determine the radius of gyration. The radius of gyration represents the distribution of mass around an axis of rotation and is used to calculate the moment of inertia.
To calculate the radius of gyration, we need the mass and length of the rod. In this case, you have mentioned that the steel rod has a mass of 500 grams and a length of 30 centimeters.
The formula to calculate the radius of gyration (k) is:
k = √(I / m),
where k is the radius of gyration, I is the moment of inertia, and m is the mass of the object.
To determine the moment of inertia (I) of a rod spinning about its axis, we use the formula:
I = (1/12) * m * L^2,
where I is the moment of inertia, m is the mass of the rod, and L is the length of the rod.
Now let's calculate the radius of gyration using these formulas:
First, convert the mass to kilograms:
m = 500 grams = 0.5 kilograms.
Next, substitute the values into the formula for moment of inertia:
I = (1/12) * 0.5 kg * (0.3 m)^2
I = (1/12) * 0.5 kg * 0.09 m^2
I = (1/12) * 0.045 kg m^2
I = 0.00375 kg m^2
Finally, substitute the moment of inertia and mass into the formula for radius of gyration:
k = √(I / m)
k = √(0.00375 kg m^2 / 0.5 kg)
k = √0.0075 m
k ≈ 0.0866 m
Therefore, the radius of gyration of the steel rod is approximately 0.0866 meters.