Conservation of energy question

In summary, a mass attached to a spring with a constant of 100 N/m and a mass of 1 kg has an initial position of 3 m and an initial velocity of 5 m/s. The period of oscillations is 0.63 s and the amplitude is 3.04 m. The amplitude is determined by the total energy of the system, which is equal to the sum of the initial potential and kinetic energies. The direction of the initial 5 m/s velocity is not needed to calculate the amplitude.
  • #1
Queequeg
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Homework Statement


A mass is attached to a spring (on a wall) of constant 100 N/m. The mass is 1 kg. The mass has an initial position of 3 m from the equilibrium position and is given an initial velocity of 5 m/s. Find the period and amplitude of oscillations.

Homework Equations


[/B]
Period: T=2(pi)(m/k)^(1/2)

Spring potential energy: (1/2)*k*x^2

Spring kinetic energy: (1/2)*m*v^2

The Attempt at a Solution



The period is the equation above, so T = 2(pi)(1/100)^(1/2)=0.63 s

I'm having a bit of trouble with the amplitude. After awhile, I realized that the potential energy is at a maximum at the amplitude and is equal to the total energy because the mass is at rest at the amplitude.

The total energy from the initial conditions is the sum of the initial potential and kinetic energies

(1/2)*100*3^2+(1/2)*1*5^2 = 462.5 J = (1/2)*k*A^2 = (1/2)*100*A^2, so A = 3.04 m.

Is that work right? Seems like the amplitude should be farther than only .04 m, but I guess that's the way the numbers work out.
 
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  • #2
Don't see anything wrong with your result: Amplitude is 3.04 m. Note that the 12.5 J from kinetic energy is only a small fraction of the 450 J from spring energy.

Isn't it nice that you don't have to ask which way the initial 5 m/s is ? (stretching further or towards equiilibrium position)
 
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1. What is conservation of energy?

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another.

2. Why is conservation of energy important?

Conservation of energy is important because it helps us understand and predict the behavior of physical systems. It also allows us to use energy efficiently and sustainably.

3. How is energy conserved?

Energy is conserved through various processes, such as energy conversion, transfer, and storage. In a closed system, the total amount of energy remains constant.

4. What are some real-life examples of conservation of energy?

Some examples of conservation of energy in everyday life include a pendulum swinging, a book falling off a shelf, and a ball rolling down a hill. In all of these cases, energy is transferred from one form to another, but the total amount remains the same.

5. Can energy be created or destroyed?

No, according to the law of conservation of energy, energy cannot be created or destroyed. It can only change forms or be transferred from one object to another.

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