Question

A 3.0 -kg mass is vibrating on a spring. It has a resonant angular speed of \(2.4 \mathrm{rad} / \mathrm{s}\) and a damping angular speed of \(0.14 \mathrm{rad} / \mathrm{s}\). If the driving force is \(2.0 \mathrm{~N}\), find the maximum amplitude if the driving angular speed is (a) \(1.2 \mathrm{rad} / \mathrm{s},\) (b) \(2.4 \mathrm{rad} / \mathrm{s}\), and \((\) c) \(4.8 \mathrm{rad} / \mathrm{s}\).

Short answer

Answer: The maximum amplitudes for the given driving angular speeds are approximately 0.187 m for 1.2 rad/s, 0.924 m for 2.4 rad/s, and 0.083 m for 4.8 rad/s.

Step-by-step solution

Identify the equation to use for maximum amplitude

The equation for the maximum amplitude of a damped harmonically driven mass on a spring is as follows: \(A_{max} = \frac{F_0}{m \sqrt{(\omega_0^2 - \omega^2)^2 + (2\beta\omega)^2}}\) Here, \(F_0\) is the driving force, \(m\) is the mass, \(\omega_0\) is the resonant angular speed, \(\omega\) is the driving angular speed, and \(\beta\) is the damping angular speed.

Plug in the given values

We have been given: - Mass, m = 3.0 kg - Resonant angular speed, \(\omega_0 = 2.4 \, \mathrm{rad/s}\) - Damping angular speed, \(\beta = 0.14 \, \mathrm{rad/s}\) - Driving force, \(F_0 = 2.0 \, \mathrm{N}\) We need to find the maximum amplitude for three driving angular speeds: 1. \(\omega = 1.2 \, \mathrm{rad/s}\) 2. \(\omega = 2.4 \, \mathrm{rad/s}\) 3. \(\omega = 4.8 \, \mathrm{rad/s}\)

Calculate the maximum amplitude for each driving angular speed

Now we can plug in the values and calculate the maximum amplitude for each driving angular speed: 1. For \(\omega = 1.2 \, \mathrm{rad/s}\): \(A_{max} = \frac{2.0}{3.0 \sqrt{({2.4}^2 - {1.2}^2)^2 + (2 \times 0.14 \times 1.2)^2}}\) \(A_{max} \approx 0.187 \, \mathrm{m}\) 2. For \(\omega = 2.4 \, \mathrm{rad/s}\): \(A_{max} = \frac{2.0}{3.0 \sqrt{({2.4}^2 - {2.4}^2)^2 + (2 \times 0.14 \times 2.4)^2}}\) \(A_{max} \approx 0.924 \, \mathrm{m}\) 3. For \(\omega = 4.8 \, \mathrm{rad/s}\): \(A_{max} = \frac{2.0}{3.0 \sqrt{({2.4}^2 - {4.8}^2)^2 + (2 \times 0.14 \times 4.8)^2}}\) \(A_{max} \approx 0.083 \, \mathrm{m}\)

Conclude the solution

Hence, the maximum amplitude for the given driving angular speeds are: 1. \(1.2 \, \mathrm{rad/s}\): \(\approx 0.187 \, \mathrm{m}\) 2. \(2.4 \, \mathrm{rad/s}\): \(\approx 0.924 \, \mathrm{m}\) 3. \(4.8 \, \mathrm{rad/s}\): \(\approx 0.083 \, \mathrm{m}\)

Key concepts

Resonant Angular Speed

In the context of damped harmonic motion, the concept of resonant angular speed is vital. This is the frequency at which a system naturally oscillates if not disturbed by any external forces. When a system is driven at its resonant angular speed, it reaches its maximum amplitude. The resonant angular speed, denoted by \( \omega_0 \), is determined using the formula \( \omega_0 = \sqrt{\frac{k}{m}} \), where \(k\) is the spring constant and \(m\) is the mass of the oscillator.

Understanding this helps us predict how much energy is needed to achieve maximum amplitude. In our exercise, we observe that the resonant angular speed is 2.4 rad/s, which signifies that the system would naturally vibrate at this speed without any added damping or driving force.

Resonance can be both beneficial and potentially destructive. The efficiency of energy transfer is highest at resonance, making it useful in amplifying signals in musical instruments and electrical circuits. However, it can also cause catastrophic failures in structures due to excessive vibrations, so considering resonance properties is crucial in engineering.

Damping Angular Speed

Damping angular speed, represented by \(\beta\), describes how quickly a system loses energy due to damping forces. These forces might include friction or air resistance acting on the system. The damping angular speed is a measure of how fast the amplitude of oscillation decreases over time. To calculate this, we use \(\beta = \frac{b}{2m}\), where \(b\) is the damping coefficient and \(m\) is the mass.

In our example with a damping angular speed of 0.14 rad/s, it tells us how significantly damping affects the motion of the mass-spring system. As the damping speed increases, the system takes less time to come to rest after being disturbed, which means the oscillations die out faster.

Damping is essential in controlling systems. While it reduces motion, preventing resonance from causing damage, it also brings stability by keeping systems like car suspensions and building structures in check during environmental disturbances like strong winds or earthquakes.

Driving Force

The driving force is an external force applied to a system to keep it oscillating at a specific angular speed. It ensures that the oscillations continue, despite natural losses due to damping. In our scenario, the driving force is 2.0 N applied to keep the 3.0 kg mass vibrating at different angular speeds for maximum amplitude.

The concept of driving force comes into play when you need to sustain motion against damping effects. The balance between the driving force and the damping force determines the steady-state amplitude of the oscillatory system.

• For slower angular speeds, such as 1.2 rad/s, the need for power from the driving force is lower, resulting in smaller amplitudes.
• At resonant speed of 2.4 rad/s, the system achieves maximum efficiency and amplitude.
• Beyond resonant speed, increasing to 4.8 rad/s, the amplitude drops as the system can no longer efficiently utilize the input energy.

This concept is visible in everyday devices like electric oscillators and pendulum clocks, where consistent external energy keeps the system in continuous, balanced motion without succumbing to damping.