91Ó°ÊÓ

When dealing with physical quantities, understanding how to perform cube and square calculations is essential. Let's explore this using the example from the exercise.

Firstly, **cubing a value** means raising it to the power of three. If we take a quantity like length, denoted by \(a = 9.7\, \mathrm{m}\), and cube it, we do the multiplication three times: \(a^3 = 9.7 \times 9.7 \times 9.7\). This gives us the volume, making our unit \(\mathrm{m}^3\).

Now, let's move on to **squaring a value**, which involves multiplying the value by itself. For instance, \(b = 4.2\, \mathrm{s}\) squared becomes \(b^2 = 4.2 \times 4.2 = 17.64\, \mathrm{s}^2\).

This manipulation of quantities is not just about numbers. It's also crucial to know how these operations affect units, transforming length in meters to cubic meters or time in seconds to square seconds. The nature of cubes and squares changes our understanding of dimensional space and time in physics.

In physics, maintaining unit consistency helps verify our calculations' correctness. When performing mathematical operations on physical quantities, observing the units ensures everything aligns and no steps are missing.

In the given problem, the expression is \(d = \frac{a^3}{cb^2}\). Let's examine the unit consistency. Start with \(a^3\), where the unit of \(a\) is \(\mathrm{m}\), cubing gives \(\mathrm{m}^3\).

Next, \(c = 69\, \mathrm{m/s}\). When combined with \(b^2 = 17.64\, \mathrm{s}^2\), the product \(cb^2 = 69 \times 17.64 = 1216.16\, \mathrm{m} \cdot \mathrm{s}\).

So, the units of the entire expression \(\frac{a^3}{cb^2}\) become \(\frac{\mathrm{m}^3}{\mathrm{m} \cdot \mathrm{s}^2} = \mathrm{m}^2 / \mathrm{s}^2\). Hence, dimensional analysis checks ensure the consistency and validity of our process.

Dividing expressions with both numerical values and units can be tricky but is crucial for solving physics problems. Let’s break down how this operates.

With the original expression \(d = \frac{a^3}{cb^2}\), you first complete all numerical calculations separately from the units. Start by dividing the calculated numbers; here, \(d = \frac{912.373}{1216.16}\). After computing this, you get approximately \(d \approx 0.750\).

Now consider the units of each part. The unit division is \(\frac{\mathrm{m}^3}{\mathrm{m} \cdot \mathrm{s}^2}\), simplifying this gives \(\mathrm{m}^2 / \mathrm{s}^2\).

This method ensures both numbers and units are handled correctly, leading to the right solution. Remember, keeping dimensional integrity is as important as the numerical answers you calculate.