Question

A 70 -gm mass is attached to the lower end of a coil spring suspended from the ceiling. The mass comes to rest in its equilibrium position, thereby stretching the spring \(5 \mathrm{~cm}\). The mass is then pulled down \(4 \mathrm{~cm}\) below its equilibrium position and released at \(t=0 .\) A damping mechanism provides a resistance numerically equal to \(280 x^{\prime}\), where \(x^{\prime}\) is the instantaneous velocity in centimeters per second. Find the displacement of the mass as a function of time.

Short answer

The displacement of the mass (x) as a function of time is given by: \(x(t) = Ae^{r_1t} + Be^{r_2t}\) where A and B are constants, which can be determined using the initial conditions \(x(0) = -4 cm\) and \(\frac{dx}{dt}(0) = 0\), and \(r_1\) and \(r_2\) are the roots of the quadratic equation \(0.07r^2 - 280r + 13.72 = 0\).

Step-by-step solution

Calculate the spring constant k

Let's first determine the spring constant (k) of the coil spring. We know the mass (m) and the extension of the spring in the equilibrium (∆x). We can use Hooke's law to find the spring constant k: Hooke's Law: \(F = -k ∆x\) The force F can be calculated using gravitational force: Gravitational force: \(F = mg\) Combining the two equations: \(-k ∆x = mg\) Substituting the given values: m=70 g, ∆x=5 cm and g = 9.8 m/s² \(-k(0.05 m) = (0.07 kg)(9.8 m/s^2)\) This allows us to solve for k: \(k = -\frac{(0.07 kg)(9.8 m/s^2)}{0.05 m} = 13.72 N/m\)

Write the equation of motion

Now, we can write the equation of motion for the damped harmonic oscillator: \(m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = 0\) where m is the mass, x is the displacement from equilibrium, c is the damping coefficient, and k is the spring constant. We know m, k, and the damping term (280x') from the given information: \(m = 0.07 kg\), \(k = 13.72 N/m\), and \(c\frac{dx}{dt} = 280\frac{dx}{dt}\) Plug in these values into the equation of motion: \((0.07 kg)\frac{d^2x}{dt^2} + 280\frac{dx}{dt} + 13.72x = 0\)

Solve the equation for displacement x(t)

To solve this second-order differential equation for the displacement x(t), we will assume a trial solution as: \(x(t) = e^{-rt}\) Taking its first and second derivatives with respect to time t: \(\frac{dx(t)}{dt} = -re^{-rt}\) \(\frac{d^2x(t)}{dt^2} = r^2e^{-rt}\) Now, substitute these derivatives back into the equation of motion: \((0.07 kg)(r^2e^{-rt}) + 280(-re^{-rt}) + 13.72(e^{-rt}) = 0\) This equation is true if: \(0.07r^2 - 280r + 13.72 = 0\) We can solve this quadratic equation for r (r1 and r2). Now, the general solution is given by: \(x(t) = Ae^{r_1t} + Be^{r_2t}\) We know the initial conditions: \(x(0) = -4 cm\) and \(\frac{dx}{dt}(0) = 0\). Using these initial conditions and plugging into the general solution, we can solve for the constants A and B. Finally, we will have the displacement x(t) as a function of time.

Key concepts

Differential Equations

Understanding differential equations is crucial when studying systems with continuous change, such as the motion of a damped harmonic oscillator. Differential equations relate a function to its derivatives, offering information on how a certain quantity changes over time.

When dealing with a harmonic oscillator, the movement can be described by a second-order differential equation. This type of equation includes the second derivative of displacement with respect to time, reflecting acceleration, which is a key component when determining the system's future behavior given its present state and rate of change.

To solve the differential equation for a damped harmonic oscillator, one typically assumes a trial solution, analyzes the characteristics of the equation, and applies initial conditions to find a specific solution that describes the system's motion. Such problems frequently involve a characteristic equation whose roots determine the nature of the system's response over time.

Hooke's Law

Hooke's Law is foundational when studying the physics of springs and elasticity. It states that the force needed to extend or compress a spring by some distance is proportional to that distance. This relationship is represented as:
\[ F = -k \Delta x \]
where \(F\) is the force applied to the spring, \(k\) is the spring constant (a measure of the spring's stiffness), and \(\Delta x\) is the displacement from the spring's equilibrium position.

In the context of the damped harmonic oscillator, this law helps us determine the restoring force exerted by the spring on the mass, a vital component in setting up the equation of motion. A precise understanding of Hooke's Law, coupled with accurate measurements, allows us to define the system's dynamism and eventually predict its motion.

Damping Coefficient

The damping coefficient, symbolized by \(c\), is a parameter that quantifies the resistance force acting against the motion of a system. In the context of a damped harmonic oscillator, this resistance force is often proportional to the velocity of the moving mass and acts in the opposite direction, hence the equation \(c\frac{dx}{dt}\).

The damping coefficient's value affects how quickly the oscillations decrease in amplitude. If \(c\) is large, the system experiences 'overdamping' and returns slowly to equilibrium without oscillating. Conversely, a small \(c\) results in 'underdamping,' where the system oscillates with gradually decreasing amplitude. Understanding the effects of different damping levels is essential in predicting and controlling the behavior of various physical systems, ranging from car suspensions to seismic buildings.

Spring-Mass System

The spring-mass system is a classical model in physics that describes the motion of a mass connected to a spring. The mass moves under the influence of the spring's restoring force, as described by Hooke's Law, and any damping forces present.

In the case of our exercise, the spring-mass system consists of a mass hanging from a coil spring attached to the ceiling, subject to gravitational forces and damping due to a mechanism providing resistance. The displacement, velocity, and acceleration of the mass can all be interrelated through differential equations. By solving these equations, we can determine the position of the mass at any point in time, thereby comprehensively understanding the dynamics of the system. Such systems are not only theoretical constructs but are widely applied in engineering, from vehicle suspensions to electronic circuit design.