Question

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 \(\omega ?\)

Short answer

When the total mass of the oscillating cart increases to 4 times its original value, the new angular frequency is 5.0 rad/s.

Step-by-step solution

Understand the given information

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.

Write the formula for the angular frequency

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.

Find the initial mass

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: \(10.0 = \sqrt{\frac{k}{m_i}}\)

Find the new mass

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

Write the formula for the new angular frequency

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

Find the relationship between the initial and new angular frequency

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: \(\frac{\omega_n}{\omega_i} = \sqrt{\frac{m_i}{m_n}}\)

Calculate the new angular frequency

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.