Question

A mass of \(0.404 \mathrm{~kg}\) is attached to a spring with a spring constant of \(204.7 \mathrm{~N} / \mathrm{m}\) and hangs at rest. The spring hangs from a piston. The piston then moves up and down, driven by a force given by \((29.4 \mathrm{~N}) \cos [(17.1 \mathrm{rad} / \mathrm{s}) t]\) a) What is the maximum displacement from its equilibrium position that the mass can reach? b) What is the maximum speed that the mass can attain in this motion?

Short answer

Based on the given information for a mass-spring system with an external force, the maximum displacement of the mass from its equilibrium position is approximately 0.337 meters, and the maximum speed that the mass can attain in this motion is approximately 5.76 meters per second.

Step-by-step solution

- Write down the given variables and the equation of motion.

We are given: Mass of the object, \(m = 0.404 kg\), Spring constant, \(k = 204.7 N/m\), External force, \(F(t) = 29.4 \cos(17.1t)\, N\) The equation of motion for a mass-spring system with an external force is: $$F(t) = m \ddot{x} + kx $$

- Rearrange the equation of motion to a differential equation in terms of x.

Given \(F(t) = 29.4 \cos(17.1t)\), we rearrange the equation of motion to: $$0.404\ddot{x} + 204.7x = 29.4\cos(17.1t) $$

- Find the displacement equation.

To solve this differential equation, we use the particular integral method by applying an oscillatory function with the same frequency as the external force: $$x(t) = A\cos(17.1t) + B\sin(17.1t)$$ Take the first derivative to find the velocity, \(v(t) = \dot{x}(t)\): $$v(t) = -17.1A\sin(17.1t) + 17.1B\cos(17.1t)$$ Take the second derivative to find the acceleration, \(\ddot{x}(t)\): $$\ddot{x}(t) = -290.41A\cos(17.1t) - 290.41B\sin(17.1t)$$ Now substitute \(x(t)\), \(v(t)\), and \(\ddot{x}(t)\) into the equation found in Step 2 and solve for \(A\) and \(B\): $$0.404(-290.41A\cos(17.1t) - 290.41B\sin(17.1t)) + 204.7(A\cos(17.1t) + B\sin(17.1t)) = 29.4\cos(17.1t)$$ Compare both sides of the equation: $$(-117.485944A + 204.7A)\cos(17.1t) + (-117.485944B + 204.7B)\sin(17.1t) = 29.4\cos(17.1t)$$ Now, solve for \(A\) and \(B\): $$A (87.214056) + 0B = 29.4$$ $$A = 29.4 / 87.214056 ≈ 0.336926$$ We can set up a similar equation for \(B\). However, since we are looking for the maximum displacement, the value of \(B\) will not impact our final answer. Hence: $$x(t) = 0.336926 \cos(17.1t)$$

- Find the maximum displacement and maximum speed.

To find the maximum displacement, we look for the maximum value of \(x(t)\): $$x_{max} = 0.336926$$ To find the maximum speed, we look for the maximum value of \(v(t)\) (already computed in step 3), using the calculated A value: $$v(t) = -17.1(0.336926)\sin(17.1t)$$ $$v_{max} = 17.1(0.336926) = 5.759{}$$ a) The maximum displacement of the mass from its equilibrium position is approximately \(0.337 m\). b) The maximum speed that the mass can attain in this motion is approximately \(5.76 m/s\).

Key concepts

Spring-Mass System

The spring-mass system is a classical physics model that represents a type of oscillatory motion. It consists of a mass attached to a spring, which can stretch or compress. When the mass is displaced from its equilibrium (rest) position and then released, it will experience a restoring force proportional to the displacement, as described by Hooke's Law:

\[ F = -kx \]
where \( F \) is the restoring force, \( k \) is the spring constant, and \( x \) is the displacement. This system is ideal for studying vibrations because it exhibits simple harmonic motion when no other forces are applied.

Damped Oscillations

However, in real-world scenarios, there are often damping forces such as friction or air resistance that cause the amplitude of the oscillation to decrease over time, leading to what is known as damped harmonic motion. It is also important to consider the effect of an external force on the system, as shown in the original exercise. The presence of a periodic external force, such as from a piston, can add complexity to the motion.

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. In the context of a spring-mass system subject to external forces, differential equations are used to describe the motion of the mass over time. Specifically, they are second-order differential equations because they involve the second derivative of the position, which represents acceleration.

Solving the Equation

Solving these equations often requires techniques from calculus, and the solutions describe the system's state at any point in time. The step-by-step solution provided in the original exercise employs such a method, where a particular solution to the differential equation is assumed and then solved for to determine constants that describe the motion under the influence of the external force.

Oscillatory Motion

Oscillatory motion is a type of repetitive movement around an equilibrium position, which can be seen in systems such as pendulums, springs, and even musical instruments.

Characteristics of Oscillatory Motion

The motion is characterized by parameters such as amplitude, frequency, and period. For a spring-mass system, the natural frequency of oscillation depends on the mass of the object and the spring constant, and is given by
\[ f = \frac{1}{2\text{\(\text{\textpi}\)}}\text{\textsqrt{}}\frac{k}{m} \]
For an undamped system, the amplitude remains constant. However, adding damping or an external force changes the motion, potentially leading to a complex oscillatory pattern as seen in the exercise with the piston-driven force. Understanding these elements of oscillatory motion is essential for analyzing and predicting the behavior of physical systems that undergo periodic motion.