Question

A mass weighing 16 lb stretches a spring 3 in. The mass is attached to a viscous damper with a damping constant of 2 Ib-sec/ft. If the mass is set in motion from its equilibrium position with a downward velocity of 3 in / sec, find its position \(u\) at any time \(t .\) Plot \(u\) versus \(t\). Determine when the mass first returns to its equilibrium. Also find the time \(\tau\) such that \(|u(t)|<0.01\) in. fir all \(t>\tau\)

Short answer

Question: A mass weighing 16 lb stretches a spring 3 in and is then set in motion from its equilibrium position with a downward velocity of 3 in/sec. A damping constant is given as 2 lb-s/ft. Find the equation of the damped harmonic oscillator for displacement and determine when the mass first returns to the equilibrium position. Additionally, find the time τ for which |u(t)|<0.01 in for all t > τ. Answer: 1) The spring constant is approximately \(k \approx 5.33\,\text{lb/in}\) or \(64\,\text{lb/ft}\). 2) The equation of motion for the damped harmonic oscillator is \(0.497 \frac{d^2 u}{dt^2} + 2 \frac{du}{dt} + 64 u = 0\). 3) To apply the initial conditions, \(u(0) = 0\) and \(\frac{du}{dt} \Big|_{t=0} = 0.25\,\text{ft/sec}\). 4) Solve the differential equation with the initial conditions and find the equation for displacement, \(u(t)\). 5) Plot \(u(t)\) vs. \(t\) and find the time when the mass first returns to the equilibrium position, either graphically or analytically. 6) Determine the time τ for which \(|u(t)|<0.01\) in for all \(t>\tau\) by solving the inequality using the given displacement equation.

Step-by-step solution

Finding Spring Constant

Note that the force exerted by a spring is given by Hooke's law, which states that force \(F\) is proportional to displacement \(u\): \(F = k u\) In this problem, we are given that a mass weighing 16 lb stretches a spring 3 in. This means that the force the spring exerts when it is stretched 3 inches is equal to the weight of the mass: \(k \times (3\,\text{in}) = 16\,\text{lb}\) Dividing both sides by 3 inches yields the spring constant, \(k\): \(k = \frac{16\,\text{lb}}{3\,\text{in}} \approx 5.33\,\text{lb/in}\)

Writing the Equation of Motion

The equation of motion for a damped harmonic oscillator is given by: \(m \frac{d^2 u}{dt^2} + c \frac{du}{dt} + k u = 0\) where \(m\) is the mass, \(c\) is the damping constant, and \(k\) is the spring constant. Our mass, \(m\), is given as 16 lb, which we need to convert to slugs, as it is a unit of mass: \(m = \frac{16\,\text{lb}}{32.2\,\text{ft/s}^2} \approx 0.497\,\text{slugs}\) The damping constant is given as \(c = 2\,\text{lb-s/ft}\). The spring constant we found in Step 1 is \(k \approx 5.33\,\text{lb/in}\), but we need to convert \(k\) to lb/ft: \(k \approx 5.33\,\text{lb/in} \times \frac{12\,\text{in}}{1\,\text{ft}} = 64\,\text{lb/ft}\) Now, we can write the equation of motion with our constants: \(0.497 \frac{d^2 u}{dt^2} + 2 \frac{du}{dt} + 64 u = 0\)

Applying Initial Conditions

The initial conditions provided are that the mass is set in motion from its equilibrium position with a downward velocity of 3 in/sec. So, \(u(0) = 0\) and \(\frac{du}{dt} \Big|_{t=0} = 3\,\text{in/sec}\). However, we need to convert this initial velocity to ft/sec: \(\frac{du}{dt} \Big|_{t=0} = 3\,\text{in/sec} \times \frac{1\,\text{ft}}{12\,\text{in}} = 0.25\,\text{ft/sec}\) Now that we have our initial conditions, we can plug them into our equation of motion and solve for the constants in the equation for \(u(t)\). This is a linear, second-order, homogeneous differential equation with constant coefficients. Solving this differential equation, typically using the characteristic equation, might be challenging. You can use online tools such as Wolfram Alpha or consult a differential equations textbook for guidance.

Plotting \(u(t)\) vs. \(t\) and Finding When the Mass Returns to Equilibrium

Once you have solved the equation of motion with initial conditions and found the equation for displacement from equilibrium \(u(t)\), you can plot \(u(t)\) vs. \(t\). You might use a tool like Desmos, GeoGebra, or Microsoft Excel to create the plot. To find the time when the mass first returns to its equilibrium position, look for the first time when \(u(t) = 0\) after the initial time. This can be done graphically, looking at the intersection of \(u(t)\) with the \(t\)-axis, or analytically, by solving \(u(t) = 0\) for \(t\).

Determining the Time \(\tau\) for Which \(|u(t)|\tau\)

To find the time \(\tau\) for which \(|u(t)|<0.01\) in for all \(t>\tau\), we should solve the inequality \(|u(t)|<0.01\,\text{in}\), keeping in mind to convert the result to ft, as our equation \(u(t)\) is in ft. This can be done graphically, finding the first time when the displacement dips below a thin band of \(\pm 0.01\) in (or \(\pm 0.000833\) ft) and stays within that range. Analytically, you will need to solve the inequality using the equation of \(u(t)\) obtained after solving the differential equation.

Key concepts

Hooke's Law

Hooke's law is a fundamental principle in physics that describes the behavior of springs. It states that the force exerted by a spring is directly proportional to the displacement from its rest position, provided the limit of elasticity is not exceeded. Mathematically, Hooke's law can be expressed as
\[ F = k u \]
where \(F\) is the force applied to the spring, \(u\) is the displacement, and \(k\) is the spring constant, which is a measure of the spring's stiffness. In practical terms, if you pull on a spring and stretch it, the spring will pull back with a force that's proportional to the stretch. This law is crucial in solving problems involving springs and harmonic motion, as it relates to the restoring force which pulls the system back toward the equilibrium. Understanding Hooke's law is essential in analyzing the behavior of a damped harmonic oscillator, since it provides the restoring force that is one part of the dynamic equation of the system.

Equation of Motion

The equation of motion for a damped harmonic oscillator is a differential equation that describes how the position of the system evolves over time. In general, this equation takes the form
\[ m \frac{d^2 u}{dt^2} + c \frac{du}{dt} + k u = 0 \]
where \(m\) is the mass of the object, \(c\) is the damping constant, \(k\) is the spring constant, and \(u\) is the displacement from equilibrium. This equation considers the inertial force (dependent on mass and acceleration), the damping force (proportional to the velocity), and the restoring force (according to Hooke's law). The combined effect of these forces describes the damped harmonic motion of the mass-spring system. The equation of motion provides a complete description of the system’s dynamics, and solving it allows us to predict the system's behavior at any given time.

Characteristic Equation

The characteristic equation is a tool used to solve linear differential equations with constant coefficients, like the equation for a damped harmonic oscillator. When faced with a second-order linear differential equation, the characteristic equation enables us to find solutions by reducing the differential equation into an algebraic one. This is achieved by substituting a trial solution into the form
\[ u(t) = e^{rt} \]
where \(r\) is a constant to be determined, into the differential equation. Doing so transforms the differential equation into an algebraic equation in terms of \(r\). The solutions to this algebraic equation, called the roots of the characteristic equation, determine whether the system's motion is overdamped, underdamped, or critically damped, and give us the general form for the solution of \(u(t)\).

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are used to model a wide range of natural phenomena, such as growth, decay, and oscillatory systems like the damped harmonic oscillator. In our context, the differential equation of motion takes into account inertia, damping, and restoring forces to describe the movement of the mass attached to the spring. Solving these equations requires integrating, which means finding the unknown function \(u(t)\) that satisfies the equation. The solution process often involves determining the characteristic equation and utilizing initial conditions to solve for specific constants within the general solution.

Initial Conditions

Initial conditions are the values that define the state of a system at the beginning of a process. In the context of differential equations, initial conditions allow us to find a specific solution to our problem by providing the necessary information to solve for arbitrary constants in the general solution. For instance, in the given exercise, the initial conditions are that the mass is released from its equilibrium position with a certain velocity. Mathematically, we specify these initial conditions as
\[ u(0) = 0, \quad \frac{du}{dt} \Bigg|_{t=0} = 0.25 \text{ ft/sec} \]
Given these initial conditions, we can proceed to solve the general solution of the differential equation to determine the specific path the mass takes during its motion.

Harmonic Motion Damping

Harmonic motion damping refers to the reduction in the amplitude of an oscillating system over time due to resistive forces, such as friction or air resistance. In the physical system we're exploring, damping is the result of the viscous damper. This damper exerts a force on the system that is proportional to the velocity and opposite to its direction, leading to energy loss in the system. This damping force changes the system’s behavior from simple harmonic motion (where the system would oscillate indefinitely) to damped harmonic motion. The degree of damping is an important factor in characterizing the system. If damping is too high, the system returns to equilibrium without oscillating; if it is too low, it oscillates many times before coming to rest. Properly analyzing and calculating the damping is crucial for understanding how long it will take for the system to stop oscillating and how fast it will reach a position close to its equilibrium.