Oscillations Spring constant Newton's second law

In summary, the problem is to derive an expression for the displacement and velocity of a mass as a function of time, given the equation m*(d2x/dt^2)+c*(dx/dt)+kx=0 and the values of m and k. The problem is to be solved for two different values of c, using Matlab. The attempted solution involves using the dsolve function, but it is not correct. The first step in solving this problem is to find the characteristic equation and its solutions.
  • #1
bfpri
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So i have the equation m*(d2x/dt^2)+c*(dx/dt)+kx=0, where d2x/dt^2 is the second derivative.

So I'm given that m=10 kg, and k=28 N/m. At time t=0 the mass is displaced to x=.18m and then released from rest. I need to derive an expression for the displacement x and the velocity v of the mass as a function of time where
a) c=3 N-s/M
b) c=50 N-s/m

Since I have to do this in matlab, I attempted to solve with dsolve and got
C10/exp((t*(c - (c^2 - 4*k*m)^(1/2)))/(2*m)) - (C10 - 9/50)/exp((t*(c + (c^2 - 4*k*m)^(1/2)))/(2*m))

clearly not right..How do i set it up correctly so i can solve for both x and v?

Thanks
 
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  • #2
bfpri said:
So i have the equation m*(d2x/dt^2)+c*(dx/dt)+kx=0, where d2x/dt^2 is the second derivative.

So I'm given that m=10 kg, and k=28 N/m. At time t=0 the mass is displaced to x=.18m and then released from rest. I need to derive an expression for the displacement x and the velocity v of the mass as a function of time where
a) c=3 N-s/M
b) c=50 N-s/m

Since I have to do this in matlab, I attempted to solve with dsolve and got
C10/exp((t*(c - (c^2 - 4*k*m)^(1/2)))/(2*m)) - (C10 - 9/50)/exp((t*(c + (c^2 - 4*k*m)^(1/2)))/(2*m))

clearly not right..How do i set it up correctly so i can solve for both x and v?
Actually, that answer is probably right, but it obscures what's going on in the problem.

What's the characteristic equation you get for that differential equation, and what are its solutions? That's a good place to start. (I'm assuming, perhaps incorrectly, that you know how to solve this differential equation. If you don't, we can back up a bit.)
 
Last edited:

What is the definition of oscillations?

Oscillations refer to the repetitive back and forth or up and down motion of an object about a fixed point or equilibrium position.

What is the spring constant and how is it related to oscillations?

The spring constant, denoted by "k", is a measure of the stiffness of a spring. It represents the force required to stretch or compress a spring by a certain distance. In oscillations, the spring constant is directly proportional to the frequency of oscillation and inversely proportional to the period of oscillation.

How does Newton's second law relate to oscillations?

Newton's second law states that the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass. In the case of oscillations, this law explains that the restoring force of a spring (due to its elasticity) is directly proportional to the displacement from equilibrium, resulting in the back and forth motion of the object.

Can the spring constant affect the amplitude of oscillations?

Yes, the spring constant can affect the amplitude of oscillations. A higher spring constant means a stiffer spring, resulting in a larger restoring force and thus a larger amplitude of oscillation. On the other hand, a lower spring constant will result in a smaller amplitude of oscillation.

How can the spring constant be calculated?

The spring constant can be calculated by dividing the applied force by the displacement of the spring from its equilibrium position. This can be represented by the equation k = F/x, where "k" is the spring constant, "F" is the applied force, and "x" is the displacement of the spring.

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