An empty cart, tied between two ideal springs, oscillates with $\omega=10.0 \mathrm{rad} / \mathrm{s} .$ A load is placed in the cart. making the total mass 4.0 times what it was before. What is the new value of \(\omega ?\)

We are given: 1. The initial angular frequency, \(\omega = 10.0\, \mathrm{rad} / \mathrm{s}\). 2. The mass becomes 4 times its original value after placing a load. We need to find the new value of \(\omega\) after the mass change.

The formula for the angular frequency of a mass-spring system is: \(\omega = \sqrt{\frac{k}{m}}\) In our case, \(k\) (the spring constant) does not change, but the mass (\(m\)) does.

Before we can find the new value of \(\omega\), let's assume the initial mass is \(m_i\), so we can write the initial angular frequency as: \(\omega_i = \sqrt{\frac{k}{m_i}}\) Given the initial angular frequency as \(\omega_i = 10.0\, \mathrm{rad} / \mathrm{s}\), we can rewrite the previous equation as: \(10.0 = \sqrt{\frac{k}{m_i}}\)

The new mass is 4 times the initial mass, so we can express it as follows: \(m_n = 4m_i\)

Now, we need to find the new angular frequency (\(\omega_n\)) when the mass is \(m_n\). We can write this new equation as: \(\omega_n = \sqrt{\frac{k}{m_n}}\)

We can find a relationship between the initial and new angular frequency by dividing the equation for \(\omega_n\) by the equation for \(\omega_i\): \(\frac{\omega_n}{\omega_i} = \sqrt{\frac{k/m_n}{k/m_i}}\) Since the spring constant \(k\) is the same for both cases, we can cancel them out: \(\frac{\omega_n}{\omega_i} = \sqrt{\frac{m_i}{m_n}}\)

Using the relationship found in the previous step, we can calculate the new angular frequency (\(\omega_n\)). We know that the initial angular frequency is \(10.0\, \mathrm{rad} / \mathrm{s}\), and the new mass (\(m_n = 4m_i\)). Therefore, we can rewrite our relationship as: \(\frac{\omega_n}{10.0} = \sqrt{\frac{m_i}{4m_i}}\) Now, solve for \(\omega_n\): \(\omega_n = 10.0 \cdot \sqrt{\frac{m_i}{4m_i}}\) As the initial mass \(m_i\) appears in both the numerator and the denominator, it cancels out: \(\omega_n = 10.0 \cdot \sqrt{\frac{1}{4}}\) Solving for the new angular frequency: \(\omega_n = 10.0 \cdot \frac{1}{2} = 5.0\, \mathrm{rad} / \mathrm{s}\) The new value of \(\omega\) is 5.0 rad/s when the total mass becomes 4 times its original value.