# Calculating Total Energy of Block-Spring System

• trickymax301
Then E = \frac 1 2 k A^2 as well.In summary, the conversation discusses the total energy of a block-spring system. The two equations for total energy, E = K + U and E = (1/2)kA^2, are both correct and equivalent. The amplitude, A, is defined as the maximum displacement and is equal to x in the first equation. The final answer for the total energy is .75 J.
trickymax301
Question: A block of mass .05 kg is pulled .3 m from its equilibrium position and released. The spring constant is 5 N/m

What is the total energy of the block-spring system?

My book says E = K + U or E = (1/2)mv^2 + (1/2)kx^2. My book also says E_total = (1/2)kA(amplitude)^2. Which formula is correct? I've just solved for the E_total equation and I found the answer to be .75 J. Can anyone confirm this answer for me? My work is below...

(1/2)(5)(.3^2) = .75 J

Thanks

How did you get .75J from that? Try again.

i lied. .225 J

Thanks for checking for me. I forgot to square the .3

With that cleared up.

The 2 relationships you have posted.

$$E = K + U = \frac 1 2 m v^2 + \frac 1 2 k x^2$$

and

$$E = \frac 1 2 k A^2$$

are the same. The amplitude is defined as when the displacement is at its maximum so A = x. At the point in time when the displacement is at its maximum, the velocity is zero. So in the first relationship, let x=A and v=0.

## 1. How do I calculate the total energy of a block-spring system?

To calculate the total energy of a block-spring system, you will need to use the formula E = 1/2kx^2, where E is the total energy, k is the spring constant, and x is the displacement of the block from its equilibrium position. This formula takes into account both the potential energy stored in the spring and the kinetic energy of the block.

## 2. What is the difference between potential energy and kinetic energy in a block-spring system?

Potential energy is the energy stored in the spring due to its displacement from its equilibrium position. This energy is converted into kinetic energy as the spring pushes or pulls the block back towards its equilibrium position. Kinetic energy is the energy of motion and is directly related to the speed of the block as it oscillates back and forth.

## 3. Can the total energy of a block-spring system ever be negative?

No, the total energy of a block-spring system cannot be negative. The potential energy stored in the spring may be negative if the spring is compressed and the block is pushed beyond its equilibrium position, but the kinetic energy will always be positive. In a conservative system like a block-spring system, the total energy will remain constant and cannot be negative.

## 4. How does the mass of the block affect the total energy of the system?

The mass of the block does not directly affect the total energy of the system. The total energy is determined by the spring constant and the displacement of the block. However, the mass of the block does affect the amplitude and period of oscillation, which in turn can affect the total energy of the system.

## 5. Can the total energy of a block-spring system change over time?

No, in a conservative system like a block-spring system, the total energy will remain constant over time. This means that the potential energy and kinetic energy will fluctuate, but their sum will always be the same. Any external factors, such as friction, may cause the total energy to decrease over time, but this is not inherent to the block-spring system itself.

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