11) Discussion Topic: 1. A mass-spring system has a mass \( (\mathrm{m}) \) of 0.5 kg and a stiffness \( (\mathrm{k}) \) of \( 100 \mathrm{~N} / \mathrm{m} \). Calculate

To calculate the properties of a mass-spring system, we can use the following formulas: 1. Natural Frequency (\( \omega_n \)): \[ \omega_n = \sqrt{\frac{k}{m}} \] 2. Period (\( T \)): \[ T = \frac{2\pi}{\omega_n} \] 3. Frequency (\( f \)): \[ f = \frac{1}{T} \] Given: - Mass (\( m \)): 0.5 kg - Stiffness (\( k \)): 100 N/m Let's calculate the natural frequency (\( \omega_n \)), period (\( T \)), and frequency (\( f \)) of the mass-spring system. Calculate the value by following steps: - step0: Calculate: \(\sqrt{\frac{100}{0.5}}\) - step1: Divide the numbers: \(\sqrt{200}\) - step2: Write as a product: \(\sqrt{100\times 2}\) - step3: Write in exponential form: \(\sqrt{10^{2}\times 2}\) - step4: Use the properties of radicals: \(\sqrt{10^{2}}\times \sqrt{2}\) - step5: Simplify the root: \(10\sqrt{2}\) The natural frequency (\( \omega_n \)) of the mass-spring system is approximately \( 14.142136 \, \text{rad/s} \). Next, let's calculate the period (\( T \)) of the mass-spring system. Calculate the value by following steps: - step0: Calculate: \(\frac{2\pi }{14.142136}\) - step1: Convert the expressions: \(\frac{2\pi }{\frac{1767767}{125000}}\) - step2: Multiply by the reciprocal: \(2\pi \times \frac{125000}{1767767}\) - step3: Multiply: \(\frac{2\pi \times 125000}{1767767}\) - step4: Multiply: \(\frac{250000\pi }{1767767}\) The period (\( T \)) of the mass-spring system is approximately 0.444288 seconds. Finally, let's calculate the frequency (\( f \)) of the mass-spring system. Calculate the value by following steps: - step0: Calculate: \(\frac{1}{0.444288}\) - step1: Convert the expressions: \(\frac{1}{\frac{6942}{15625}}\) - step2: Multiply by the reciprocal: \(\frac{15625}{6942}\) The frequency (\( f \)) of the mass-spring system is approximately 2.250792 Hz. Therefore, the properties of the mass-spring system are: - Natural Frequency (\( \omega_n \)): \( 14.142136 \, \text{rad/s} \) - Period (\( T \)): \( 0.444288 \, \text{s} \) - Frequency (\( f \)): \( 2.250792 \, \text{Hz} \)