For the spring-mass system shown, determine the static deflection δst, the system period τ, and

For the spring-mass system shown, determine the static...

Key Concepts

Static Equilibrium

Static equilibrium in a spring-mass system is reached when the forces acting on the mass balance each other so that there is no net acceleration. In this context, the static deflection is the displacement from the unstressed position where the gravitational force is exactly balanced by the spring’s restoring force (mg = k?). This concept is essential as it defines the new equilibrium position around which oscillatory motion occurs.

System Period (Natural Period)

The system period, often referred to as the natural period, is the time it takes for the mass-spring system to complete one full cycle of oscillation when displaced from equilibrium. It is determined by the mass and the spring constant, typically expressed by the formula ? = 2??(m/k) in simple harmonic motion. The period is a key characteristic because it governs the timing of the oscillatory response of the system.

Maximum Velocity in Oscillatory Motion

In a mass-spring system undergoing simple harmonic motion, the maximum velocity is attained as the mass passes through the equilibrium position where all the potential energy has been converted to kinetic energy. The velocity can be derived from energy conservation principles, reflecting the balance between kinetic and potential energy changes during the oscillation. Understanding this maximum velocity is crucial for analyzing the dynamic response and the energy distribution within the system.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes the type of periodic oscillation experienced by an ideal spring-mass system without damping and external forcing. In SHM, the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. This concept underpins the analysis of the system’s behavior, including the calculation of static deflection, period, and maximum velocity, by providing a theoretical framework based on sinusoidal motion.

Transcript

00:01 Hello, today we're going to answer a problem related to the several parameters regarding this image on the left, where we have a mass vibrating on some spring with spring constant k.

00:14 The values of the spring constant and the mass are given, and we are told that the mass starts at rest, but has an initial displacement of 100 millimeters in this particular problem, but the value could be anything.

00:28 What we're looking for are these three parameters, the maximum deflection, the period, and the velocity maximum during this oscillation.

00:40 So we can do some of, many of these are just simple plug -in chug.

00:45 The relationships are derived in the book.

00:47 The value for this maximum deflection is the weight divided by the spring constant, and if we plug in our mass, times gravity divided by our spring constant where everything is already in our proper si units we get here 0 .2 meters pretty simple on that guy next we need to find the period tau now if you recall the period is one over the frequency and the frequency is the equation that we're typically used to seeing we're going to write that as the inverse so instead of 1 over 2 pi we're going to write 2 pi and instead of k over m we're going to write m over k and here again we know the mass we know the spring constant if you plug in those parameters so 2 pi square root of 2 over 98 and you multiply that out you end up with 0 .89 seconds so again not too bad here the last portion the v max is a little more complicated so if we go back to recall the general equation for the position of this oscillator, an undamped oscillator, as a function of time.

02:04 We have this format plus an x -not dot over omega -sign omega -t...

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