Period, amplitude, maximum speed, and total energy

In summary, to determine the period, amplitude, maximum speed, and total energy of a 175g mass attached to a horizontal spring oscillating at a frequency of 2.0Hz, we can use the equations for mass-spring frequency, total energy, and maximum speed and amplitude. The period is 0.50s, the amplitude is 5.25cm, the maximum speed is 65.9cm/s, and the total energy is 0.0380J.
  • #1
aligass2004
236
0

Homework Statement


A 175g mass attached to a horizontal spring oscillates at a frequency of 2.0Hz. At one instant, the mass is at x = 5.0cm and has Vx = -20cm/s. Determine the following.

a.) the period
b.) the amplitude
c.)the maximum speed
d.)the total energy


Homework Equations





The Attempt at a Solution


I solved part a by using T = 1/f = .5s. I'm not sure how I find the amplitude though. To find the maximum speed, I think I would just use Vmax = 2(pi)fA or (2 pi A)/T. I'm also not sure how I would find the total energy.
 
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  • #2
A complete solution is offered:

Let's take the parts in reverse order, beginning with the energy.

Part (d): The total energy of the system
We first determine the spring constant using the mass-spring frequency equation:

## f = \frac{1}{2 \pi} \sqrt{\frac{k}{M}}##

## k = 4 \pi^2 f^2 M = 27.64~N/m##

The total energy of the system comprises kinetic and potential energy associated with the mass and the spring respectively.

##E = \frac{1}{2} M v^2 + \frac{1}{2} k x^2##

where v is the velocity of the mass and x the displacement from equilibrium (stretch or compression of the spring). We are given the velocity and position (presumably a displacement from equilibrium) at a particular instant.

##M = 175~gm = 0.175~kg##
##v = -20~cm/s = -0.20~m/s##
##x = 5.0~cm = 0.050~m ##

So that:

##E = \frac{1}{2} (0.175~kg) (-0.20~m/s)^2 + \frac{1}{2} (27.64~N/m)(0.050~m)^2##
##E = 0.0380~J##

Part (c): The maximum speed
We can use the expression for the total energy. The speed will be maximum when all the energy is due to the speed of the mass and the spring is at its equilibrium position with no stored energy. So:

##E = \frac{1}{2} M V_{max}^2##

##V_{max} = \sqrt{\frac{2E}{M}} = 65.9~cm/s##

Part(b): The amplitude
Again using the energy expression, the maximum amplitude occurs when the mass is stationary and the spring is at its maximum extension, so all the system energy is in the spring. So:

##E = \frac{1}{2} k x_{max}^2##

##x_{max} = \sqrt{\frac{2E}{k}} = 5.25~cm##

Part(a) The period of oscillation
This is simply the inverse of the frequency (2.0 Hz), so

##T = \frac{1}{f} = 0.50~s##
 

1. What is the period of a wave?

The period of a wave is the amount of time it takes for one complete cycle or oscillation to occur. It is usually measured in seconds, and is inversely proportional to the frequency of the wave.

2. How is amplitude defined?

Amplitude is the maximum displacement or distance of a wave from its equilibrium or rest position. It is a measure of the intensity or strength of the wave, and is usually measured in meters.

3. What is the maximum speed of a wave?

The maximum speed of a wave is the highest velocity that a particle in the wave can reach. This usually occurs at the peak or trough of the wave and is dependent on the amplitude and frequency of the wave.

4. How is total energy related to period and amplitude?

The total energy of a wave is directly proportional to its amplitude and frequency, and indirectly proportional to its period. This means that a wave with a higher amplitude and frequency will have more energy, while a wave with a longer period will have less energy.

5. Can the total energy of a wave change?

Yes, the total energy of a wave can change due to factors such as changes in amplitude or frequency, interference with other waves, or absorption or reflection by objects. However, the total energy of a closed system will always remain constant.

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