Amplitude and Total Energy

In summary, amplitude is a measure of the maximum displacement of a wave and is directly related to its energy. It can change depending on the medium and energy of the wave. Total energy can be calculated by multiplying the square of amplitude with a constant. When the amplitude is doubled, the total energy increases by a factor of four, and when halved, it decreases by a factor of four.
  • #1
hangten1039
17
0
A mass of 0.5 kg is attached to the end of a massless spring of spring constant 0.40 N/m. It is released from rest from an extended position. After 0.7 s, the speed of the mass is measured to be 1.75 m/s. What is the amplitude of oscillation? What is the total energy (relative to the mass at rest in the unextended position) contained in this system?



x=Acoswt
conservation of energy



I've tried many different attempts.
 
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  • #2
Anyone have any help on this problem. I think I have to use conservation of energy but I really am not sure
 
  • #3
But I'm getting a bit confused. Can you please help?

Sure, let's break down the problem step by step.

First, we can use the equation for simple harmonic motion to find the amplitude (A) of the oscillation. In this case, the mass (m) is 0.5 kg and the spring constant (k) is 0.40 N/m. We also know that the speed (v) of the mass after 0.7 seconds is 1.75 m/s. The equation for simple harmonic motion is:

v = Aωcos(ωt)

Where ω is the angular frequency, which is equal to √(k/m). Plugging in the values, we get:

1.75 = A√(0.40/0.5)cos(√(0.40/0.5)*0.7)

Solving for A, we get A = 0.7 m. So, the amplitude of oscillation is 0.7 meters.

Next, we can use the conservation of energy to find the total energy contained in the system. The total energy (E) is equal to the sum of the kinetic energy (KE) and the potential energy (PE). The equation for total energy is:

E = KE + PE

Kinetic energy is equal to 1/2 * m * v^2, and potential energy is equal to 1/2 * k * x^2. In this case, the mass is moving with a speed of 1.75 m/s and the spring is extended by 0.7 m. So, we can plug in these values to get:

KE = 1/2 * 0.5 * (1.75)^2 = 0.76625 J

PE = 1/2 * 0.40 * (0.7)^2 = 0.098 J

Therefore, the total energy in the system is 0.76625 J + 0.098 J = 0.86425 J. This is the energy relative to the mass at rest in the unextended position.

I hope this helps clarify the problem for you. Let me know if you have any other questions.
 

1. What is amplitude?

Amplitude is a measure of the maximum displacement of a wave from its rest position. It is typically represented by the height of the wave on a graph and is directly related to the wave's energy.

2. How is amplitude related to total energy?

The amplitude of a wave is directly proportional to its total energy. This means that as the amplitude increases, so does the total energy of the wave. This relationship is described by the formula E = kA^2, where E is energy, A is amplitude, and k is a constant.

3. Can the amplitude of a wave change?

Yes, the amplitude of a wave can change depending on the medium it is traveling through and the amount of energy it carries. In some cases, a wave's amplitude may decrease due to factors such as friction or damping, resulting in a decrease in total energy.

4. How is total energy calculated?

Total energy (E) can be calculated by multiplying the square of the amplitude (A) by a constant (k), as described by the formula E = kA^2. The value of the constant depends on the type of wave and the medium it is traveling through.

5. What happens to total energy when the amplitude is doubled?

If the amplitude of a wave is doubled, the total energy will increase by a factor of four. This is because the energy is directly proportional to the square of the amplitude. Similarly, if the amplitude is halved, the total energy will decrease by a factor of four.

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