Question

Verify that \(\frac{1}{2} I \omega^{2}\) has dimensions of energy.

Short answer

Answer: The dimensions of the formula \(\frac{1}{2} I \omega^{2}\) are (M × L² × T⁻²), which represent dimensions of energy.

Step-by-step solution

Identify dimensions of each variable

In the given formula \(\frac{1}{2} I \omega^{2}\), we have two variables, I and ω. I represents the moment of inertia, which has dimensions of mass (M) times distance squared (L²). ω represents angular velocity, which has dimensions of inverse time (T⁻¹).

Determine the dimensions of the formula

Now we need to determine the dimensions of the formula itself. In order to do this, we should determine the dimensions of \(\omega^{2}\) first: The dimensions of \(\omega^{2}\) would be (T⁻¹)² = T⁻². Now, we can find the dimensions of the entire formula by multiplying the dimensions of I and \(\omega^{2}\): (M × L²) × (T⁻²).

Compare dimensions with dimensions of energy

Finally, we need to compare the dimensions of our formula with the dimensions of energy. The dimensions of energy are mass (M) times distance squared (L²) divided by time squared (T²), or (M × L² × T⁻²). Comparing the dimensions of our formula and energy, we see that they are the same: (M × L² × T⁻²). Thus, we can conclude that the formula \(\frac{1}{2} I \omega^{2}\) has dimensions of energy.