Total energy of mass spring system?

In summary, the conversation discusses finding the general energy of an object with a given mass and equation of motion. The suggested equations for finding the energy are E=mv^2/2 +kA^2/2 and E=(1/2)kx^2 + (1/2)mv^2, where x is the position at a given time and v is the speed at that time. It is suggested to use either the kinetic energy or the elastic potential energy, depending on the given conditions, to find the total energy of the system. It is also mentioned that if the object is oscillating vertically, gravitational potential energy should also be considered.
  • #1
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Homework Statement



We have an object whose mass is 0.7 kg moves with an equation y=0.45cos8.4t.
Find the general energy

w=8.4 A=0.45

Homework Equations



E=mv^2/2 +kA^2/2

The Attempt at a Solution



So I found k using w=(k/m)^0.5. And then I found V0 using V0=A*(k/m)^0.5
and then i found x using F=kx
Then I found V=V0*(1-x^2/A^2)^0.5
And then I used E=kA^2/2+mv^2/2

IS THIS RIGHT?
 
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  • #2
When the object is at 0 displacement and max speed v0, all of its energy is kinetic. Therefore, you can find the total energy of the system by just computing the kinetic energy (1/2)mv0^2 at this instant.

Alternatively, a quarter of a period later, when the object is at max displacement A and 0 speed, all of the kinetic energy that it had has been converted into elastic potential energy in the spring. Therefore, at this instant, you can compute the total energy of the system simply by computing the elastic potential energy (1/2)kA^2

Both of these expressions will give you the same answer for the total energy of the system, so you would use one, or the other, but not both.

The expression for the total system energy at an arbitrary time t is just the sum of the kinetic and potential energy of the mass:

E = (1/2)kx^2 + (1/2)mv^2

Where "x" is the position at time t, and "v" is the speed at time t. However, it is easiest to pick a time t where one of these two energy terms is zero, like I did in the two cases above. The first case was for x=0, v=v0. The second case was for x=A, v=0. Do you understand now?

One more thing. I notice that you used "y" instead of "x" to denote the position of the mass. This suggests to me that the mass is oscillating vertically. If that's true, then you need to consider *gravitational* potential energy as well.
 
  • #3
Thank you so much.Now I understand it all :)
So I guess that now I only have to use w=(k/m)^0.5 and F=kx to find x :)
 
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  • #4
Elaia06 said:
Thank you so much.Now I understand it all :)
So I guess that now I only have to use w=(k/m)^0.5 and F=kx to find x :)

No, you don't need to find "x" (or "y" in this case). Like I clearly explained above, once you know k and A, you can get the total energy of the system.


OR, once you know m and v0, you can get the total energy of the system.
 
  • #5


I must say that your approach is not entirely correct. The total energy of a mass-spring system is given by the sum of the kinetic energy (KE) and potential energy (PE). The equation you have used, E=mv^2/2 +kA^2/2, is the total energy equation for a simple harmonic oscillator, where m is the mass, v is the velocity, k is the spring constant, and A is the amplitude of oscillation. However, in this problem, we are given an equation for the displacement of the object, not its velocity. So, we cannot directly use this equation.

To find the total energy of the system, we need to use the equation for PE, which is given by PE = 1/2*k*x^2, where x is the displacement of the object from its equilibrium position. In this case, the equation for displacement is y=0.45cos8.4t, so we can find the displacement at any given time t.

Therefore, the general energy of the system can be expressed as:

E = PE + KE = 1/2*k*x^2 + 1/2*m*v^2

where x = 0.45cos8.4t and v = -0.45*8.4*sin8.4t (since v=dy/dt).

To find the value of k, we can use the given values of w and A, as you have correctly done. So, k = m*w^2/A^2 = 0.7*(8.4)^2/(0.45)^2 = 216.89 N/m.

Now, we can substitute the values of x and v in the equation for energy to get the general expression:

E = 1/2*216.89*(0.45cos8.4t)^2 + 1/2*0.7*(-0.45*8.4sin8.4t)^2

= 48.61cos^2(8.4t) + 0.662sin^2(8.4t) Joules

So, the general energy of the system is given by the above expression, which will vary with time. I hope this helps to clarify your approach and gives you a better understanding of the concept of energy in a mass-spring system.
 

1. What is the equation for calculating the total energy of a mass spring system?

The total energy of a mass spring system can be calculated using the equation E = 1/2kx^2 + 1/2mv^2, where k is the spring constant, x is the displacement from equilibrium, and m is the mass of the object on the spring.

2. How does the total energy change as the mass and spring constant of the system are varied?

The total energy of a mass spring system is directly proportional to the square of the displacement and the spring constant, and inversely proportional to the mass. This means that as the mass or spring constant increases, the total energy will also increase. However, as the mass decreases, the total energy will decrease.

3. Can the total energy of a mass spring system be negative?

Yes, the total energy of a mass spring system can be negative. This occurs when the object on the spring is displaced in the opposite direction of the spring force, resulting in negative potential energy. However, the total energy of the system will always remain constant.

4. What factors can affect the total energy of a mass spring system?

The total energy of a mass spring system can be affected by several factors, including the mass of the object, the spring constant, the initial displacement, and external forces such as friction or air resistance.

5. What is the significance of the total energy in a mass spring system?

The total energy of a mass spring system represents the sum of its kinetic and potential energy. This energy is conserved and can be converted between kinetic and potential energy as the object oscillates between its equilibrium position and maximum displacement. Understanding the total energy of a mass spring system is crucial in analyzing and predicting its behavior.

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