Question

Show that the equation \(a=-\omega^{2} x\) is consistent for units, and that \(\sqrt{k / m}\) has the same units as \(\omega\).

Short answer

Answer: Yes, the equation \(a = -\omega^2 x\) is consistent in terms of units as both sides of the equation have units \(\frac{m}{s^2}\). Furthermore, the units of \(\sqrt{k/m}\) match the units of \(\omega\) as both have units of \(\frac{rad}{s}\).

Step-by-step solution

Analyze the given equation

We are given the equation \(a = -\omega^2 x\), where \(a\) is the acceleration, \(\omega\) is the angular frequency, and \(x\) is the position. We need to show that this equation is consistent in terms of units.

Check units for the left side of the equation

Let's check the units for the left side of the equation, which is the acceleration \(a\). The unit for acceleration is \(\frac{m}{s^2}\) (meters per second squared).

Check units for the right side of the equation

Now let's check the units for the right side of the equation (\(-\omega^2 x\)). The unit for angular frequency (\(\omega\)) is rad/s (radians per second), and the unit for position (\(x\)) is meters (m). The right side of the equation becomes \(\frac{rad^2}{s^2} \cdot m\).

Verify that units are consistent

Since both left and right sides of the equation have units \(\frac{m}{s^2}\), we can conclude that the equation is consistent in terms of units.

Analyze the expression for angular frequency

We are given the expression \(\sqrt{k/m}\) and need to show that it has the same units as \(\omega\). Let's analyze the units in the expression.

Check units in the expression

We know that \(k\) represents the spring constant with units N/m (Newtons per meter) and \(m\) represents the mass with units kg (kilograms). Since \(\omega\) has units rad/s, we need to show that the expression \(\sqrt{k/m}\) has the same units.

Find the units of the expression

Let's find the units of the expression: \(\sqrt{\frac{N/m}{kg}} = \sqrt{\frac{kg\cdot m/s^2}{kg}} = \sqrt{\frac{m}{s^2}}\) Notice that we multiplied the numerator by the conversion factor \(\frac{1N}{1kg \cdot m/s^2}\) to obtain the units in terms of mass, length, and time.

Compare the units of the angular frequency and the expression

Now we can see that the units of the expression \(\sqrt{k/m}\) are \(\frac{rad}{s}\) (since \(\sqrt{\frac{m}{s^2}}\) in the numerator represents radians). This is the same as the units of \(\omega\) (i.e., rad/s). We have now shown that the equation \(a = -\omega^2 x\) is consistent for units, and that \(\sqrt{k/m}\) has the same units as \(\omega\).