# Dynamics: Spring problem (Oscillation)

• Dean-o
In summary, the mass of 4 kg and spring constant of 64 N/m are provided, with the spring being unstretched at x = 0. The mass has a velocity of 2 m/s down an inclined surface at t = 0, with an angle of incline of 20 degrees and a spring attached on its left surface. The equation for the system is given as x = Asin(wt)+Bcos(wt), with the solution including an additional value of 0.210 m, which represents the equilibrium position required to balance the downslope gravity.
Dean-o

## Homework Statement

The mass m = 4 kg and the spring constant k = 64 N/m. The spring is unstretched when x = 0.
At t = 0, x = 0 and the mass has a velocity of 2 m/s down the inclined surface. What is the value
of x at t = 0.8 s?
The angle of incline is 20 degrees, and with the mass moving down to the right with a spring attached on its left surface.

## Homework Equations

ΣFx = max
x = Asin(wt)+Bcos(wt)

## The Attempt at a Solution

d2x/dt2 + (16 s-2)x = 3.355 m/s2

I got to this point and I found that for the equation x = Asin(wt)+Bcos(wt) the solution gives x = Asin(wt)+Bcos(wt) + 0.210 m. I get that 0.210 = 3.355/16, but why is it included in this solution for the equation when all other examples of this problem for a vertically or horizontally held spring mass system use just the x = Asin(wt)+Bcos(wt) equation without this addition value? What is this value, and is it just disregarded as zero in the other forms of this problem?

Dean-o said:
What is this value
It is the equilibrium position, i.e. the spring extension required to balance the downslope gravity.

Dean-o
haruspex said:
It is the equilibrium position, i.e. the spring extension required to balance the downslope gravity.

Oh okay, that makes sense, thanks!

## 1. What is a spring problem in dynamics?

A spring problem in dynamics refers to the study of the motion of a mass attached to a spring that is subject to external forces. This type of problem is often used to understand the behavior of mechanical systems, such as in engineering and physics.

## 2. What is oscillation in dynamics?

Oscillation in dynamics is the repetitive back-and-forth motion of a system around an equilibrium point. This type of motion is often seen in systems with a restoring force, such as a mass-spring system, and can be described using concepts such as frequency, amplitude, and period.

## 3. How is the period of oscillation determined in a spring problem?

The period of oscillation in a spring problem is determined by the mass of the object attached to the spring and the stiffness of the spring itself. The period can be calculated using the formula T = 2π√(m/k), where T is the period, m is the mass, and k is the spring constant.

## 4. What is the relationship between amplitude and energy in a spring problem?

In a spring problem, the amplitude of oscillation is directly proportional to the energy of the system. This means that as the amplitude increases, so does the energy of the system. The energy is highest at the maximum displacement and lowest at the equilibrium point.

## 5. How does damping affect the motion of a spring problem?

Damping in a spring problem refers to the gradual decrease in amplitude of the oscillation over time. This can be caused by external forces, such as friction, and can result in the system reaching equilibrium faster. Damping can also change the period of oscillation and reduce the maximum displacement of the mass.

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