Question

A single degree of freedom system is represented as a \(4 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(6 \mathrm{~N} / \mathrm{m}\). If the coefficients of static and kinetic friction between the mass and the surface it moves on are \(\mu_{s}=\mu_{k}=0.1\), and the mass is displaced 2 meters to the right and released with a velocity of \(4 \mathrm{~m} / \mathrm{sec}\), determine the time after release at which the mass sticks and the corresponding displacement of the mass.

Short answer

Answer: To find the time when the mass sticks and the displacement at that time, follow these steps: 1. Calculate the maximum static friction force using the formula: \(F_{f_{max}} = \mu_s \cdot mg\). 2. Calculate the maximum spring force to make the mass stick by equating it with the maximum static friction force: \(F_s = F_{f_{max}}\). 3. Find the displacement that corresponds to the sticking condition by solving for \(x\): \(x = \frac{\mu_s \cdot mg}{k}\). 4. Calculate the displacement when the mass sticks, using the given values and the formula obtained in step 3. 5. Calculate the time when the mass sticks, using the position equation of a mass oscillating in a spring system: \(x(t) = A \cos(\omega t) + \frac{v_0}{\omega} \sin(\omega t)\), and solve for time \(t\) when the mass sticks by substituting the calculated displacement. After following these steps, you will obtain both the time when the mass sticks due to friction and the corresponding displacement.

Step-by-step solution

Calculate the maximum static friction force

Since the coefficients of static and kinetic friction are the same, we can simply calculate the maximum static friction force acting on the mass when it sticks, which will be given by \(F_{f_{max}} = \mu_s \cdot N = \mu_s \cdot mg\), where \(\mu_s = 0.1\) is the coefficient of static friction, \(m = 4 \, \mathrm{kg}\) is the mass of the object, and \(g = 9.81 \, \mathrm{m/s^2}\) is the acceleration due to gravity.

Calculate the maximum spring force to make the mass stick

When the mass sticks, the spring force will be equal to the maximum static friction force. The spring force is given by \(F_s = k \cdot x\), where \(k = 6 \, \mathrm{N/m}\) is the stiffness of the spring, and \(x\) is the displacement from the equilibrium position. To find the maximum spring force, we equate \(F_s\) with \(F_{f_{max}}\). \(F_s = F_{f_{max}}\)

Find the displacement that corresponds to the sticking condition

By equating the forces in Step 2, we can find the displacement required for the mass to stick: \(k \cdot x = \mu_s \cdot mg\), Solving for \(x\), we get: \(x = \frac{\mu_s \cdot mg}{k}\)

Calculate the displacement when the mass sticks

Substitute the given values into the formula obtained in Step 3: \(x = \frac{0.1 \cdot (4\, \mathrm{kg}) \cdot (9.81\, \mathrm{m/s^2})}{6\, \mathrm{N/m}}\) Evaluating the expression gives the displacement when the mass sticks.

Calculate the time when the mass sticks

Since the mass is released with a velocity of 4 m/s, we can find the time it takes for the mass to stick using the position equation of a mass oscillating in a spring system: \(x(t) = A \cos(\omega t) + \frac{v_0}{\omega} \sin(\omega t)\), where \(A = 2\, \mathrm{m}\) is the amplitude, \(v_0 = 4 \,\mathrm{m/s}\) is the initial velocity, \(\omega = \sqrt{\frac{k}{m}}\) is the angular frequency. By substituting the calculated displacement \(x\) for the position function, we can solve for time \(t\) when the mass sticks. Once we have the time when the mass sticks, and the corresponding displacement, we have our solution for this exercise.

Key concepts

Understanding Single Degree of Freedom Systems

A single degree of freedom (SDOF) system is one of the simplest models used in the study of vibrations, representing the minimum number of independent coordinates necessary to describe the motion of a system. An SDOF system has a mass that moves in one direction only, such as back and forth along a straight line. In the provided exercise, we see an example of an SDOF system consisting of a 4 kg mass attached to a spring.

When analyzing an SDOF, we often consider its natural frequency, which depends on the mass involved and the stiffness of the spring – depicted mathematically by the equation \(\omega = \sqrt{\frac{k}{m}}\). This natural frequency plays a crucial role in understanding how the system will respond to different types of inputs, such as external forces or initial disturbances.

The Role of Static and Kinetic Friction in Vibrations

Friction is a resistant force that acts opposite to the relative motion between two surfaces in contact. The two main types of friction involved in engineering vibrations are static and kinetic friction. Static friction, \(\mu_s\), occurs when the surfaces are not moving relative to each other and is generally higher than kinetic friction, \(\mu_k\), which is present when there is relative motion.

In our exercise, both static and kinetic friction coefficients are given as 0.1, simplifying calculations since the transition between sticking and sliding will not be due to a change in the frictional coefficient but due to the applied forces. The static frictional force is critical to determine when the mass will no longer oscillate and instead remain at rest, sticking to the surface. The equation \(F_{f_{max}} = \mu_s \cdot mg\) represents the maximum force of static friction before slipping into motion. Understanding friction is crucial for accurately predicting system behavior, as it has a significant impact on the system's energy dissipation and motion characteristics.

Spring System Oscillation and Damping

Spring system oscillation is a classic mechanical phenomenon in which a mass attached to a spring moves back and forth about its equilibrium position. The motion can be characterized as oscillatory when the mass is subject only to the spring's restaurative force and potentially friction or damping forces. In an ideal system without damping, the motion would continue indefinitely. However, in practical systems like the one in our exercise, frictional forces dissipate energy, which dampens the oscillation over time.

When the damping force, in this case introduced via friction, becomes large enough to counteract the maximum restoring force the spring can produce, the system will stop oscillating and the mass will stick. Using the condition that the spring force (\(kx\)) equals the maximum static friction force (\(F_{f_{max}}\)), one can find the critical displacement that leads to the cessation of motion. Students must pay attention to this concept as understanding how spring oscillations are influenced by frictional forces helps solve a variety of practical engineering problems related to vibrations across diverse fields.