Question

These exercises deal with undamped vibrations of a spring-mass system, $$m{y}^{\prime \prime }+ky=0,y(0)=y_0,y^{\prime }(0)=y_0^{\prime }.$$ Use a value of $9.8\mathrm{m}/{\mathrm{s}}^2$ or $32\mathrm{ft}/{\sec }^2$ for the acceleration due to gravity. A 9-lb weight, suspended from a spring having spring constant $k=32\mathrm{lb}/\mathrm{ft}$, is perturbed from its equilibrium state with a certain upward initial velocity. The amplitude of the resulting vibrations is observed to be 4 in. (a) What is the initial velocity? (b) What are the period and frequency of the vibrations?

Short answer

Question: Calculate the initial velocity, period, and frequency of the vibrations for a 9-lb weight attached to an undamped spring-mass system with an amplitude of 4 inches and a spring constant of 32 lb/ft. Answer: (a) The initial velocity of the 9-lb weight is: $$v_0=0.333\cdot \sqrt{\frac{32}{\frac{9}{32}}}$$ (b) The period and frequency of the vibrations are: $$T=\frac{2\pi }{\omega }$$ $$f=\frac{\omega }{2\pi }$$

Step-by-step solution

Convert weight to mass

We are given that the weight is 9 lbs. We need to convert this weight into mass. We can use the following formula to convert weight to mass: $$m=\frac{W}{g}$$ where $W$ is the weight (in lbs), $g$ is the acceleration due to gravity (in ft/s²), and $m$ is the mass (in slugs). Using the given value of 32 ft/s² for the acceleration due to gravity, we can calculate the mass: $$m=\frac{9}{32}$$

Calculate the angular frequency

The spring equation for an undamped system is given by: $$m{y}^{\prime \prime }+ky=0$$ Let us denote the angular frequency by $\omega$. The equation can be rewritten as: $$\frac{k}{m}={\omega }^2$$ Solving for $\omega$, we get: $$\omega =\sqrt{\frac{k}{m}}=\sqrt{\frac{32}{\frac{9}{32}}}$$

Calculate the initial velocity

We are given that the amplitude of the vibration is 4 inches (0.333 ft). The amplitude can be related to the initial velocity through the equation: $$A=\frac{v_0}{\omega }$$ Rearranging the equation to solve for the initial velocity $v_0$, we get: $$v_0=A\cdot \omega =0.333\cdot \sqrt{\frac{32}{\frac{9}{32}}}$$

Calculate the period and frequency

The period ($T$) and frequency ($f$) of the vibrations can be calculated using the angular frequency. The period is the inverse of the angular frequency, and the frequency is the inverse of the period: $$T=\frac{2\pi }{\omega }$$ $$f=\frac{1}{T}=\frac{\omega }{2\pi }$$ Substitute the value of $\omega$ in the equations to find the period and frequency of the vibrations.

Answers

(a) The initial velocity of the 9-lb weight is: $$v_0=0.333\cdot \sqrt{\frac{32}{\frac{9}{32}}}$$ (b) The period and frequency of the vibrations are: $$T=\frac{2\pi }{\omega }$$ $$f=\frac{\omega }{2\pi }$$

Key concepts

Undamped Vibrations

When a mass attached to a spring is set into motion, it creates a type of oscillatory motion known as undamped vibrations. These occur when there is no external force or friction to slow down the motion, meaning the system can theoretically vibrate indefinitely. In the context of the spring-mass system, two essential conditions characterize undamped vibrations: the system is isolated from its environment and doesn't lose energy over time, and the restoring force is directly proportional to the displacement from the equilibrium position.

The absence of damping means that the total mechanical energy of the system, which is the sum of kinetic and potential energies, stays constant. The energy periodically converts from potential to kinetic and back, as the object moves through its equilibrium position and then slows down and turns back once it reaches the maximum extension or compression of the spring. Understanding undamped vibrations is crucial, as it forms a foundational concept for more complex topics in mechanics and waves.

Differential Equations

Differential equations are mathematical tools that relate a function to its derivatives, providing a way to describe change. They are widely used in physics, engineering, and mathematics to model how physical quantities evolve over time or space. A differential equation expresses the rate of change of a variable in terms of the variable itself and its derivatives.

In the exercise provided, the equation $m{y}^{″}+ky=0$ is a second-order linear homogeneous differential equation. It models the undamped vibrations of a spring-mass system. The equation implies that the acceleration of the mass $y^{″}$ (second derivative with respect to time) is proportional to its position $y$ but in the opposite direction, with the constant of proportionality being $\frac{k}{m}$. The solution to this equation involves trigonometric functions that describe the position of the mass as a function of time.

Angular Frequency

Angular frequency $\omega$ is a measure of how quickly a system undergoes rotational or oscillatory motion. In the context of a spring-mass system, it represents how rapidly the mass oscillates about its equilibrium position. It is closely related to the system's natural frequency, which is the frequency at which the system would vibrate if not subjected to damping or external forces.

The angular frequency is determined by the mass of the object and the spring constant, as shown in the formula$\omega =\sqrt{\frac{k}{m}}$. In simple terms, a stiffer spring or a lighter mass leads to a higher angular frequency. This concept is pivotal for understanding vibrational motion, as it connects the physical properties of the system to the motion's characteristics. It also plays a central role in calculating the period and frequency of the vibrations.

Period and Frequency of Vibrations

The period $T$ of a vibration is the time it takes to complete one oscillation cycle. It's an essential concept when analyzing periodic motion because it reflects how long a single back-and-forth movement takes. In contrast, the frequency $f$ is the number of oscillations that occur in a unit of time, often one second, and is the reciprocal of the period.To calculate these values from angular frequency, you can use these relationships:$T=\frac{2\pi }{\omega }$ and$f=\frac{1}{T}=\frac{\omega }{2\pi }$.Likewise, knowing the period or frequency can help determine other properties of the system, like angular frequency. These concepts help us describe and predict how the spring-mass system or any harmonic oscillator behaves over time.