Calculating Total Energy in Simple Harmonic Motion

[Lavace]

Homework Statement

Calculate the total energy of a 2kg mass that is undergoing simple harmonic motion with an amplitude of 1cm and and frequency of 16Hz.

Homework Equations

ma = -w^{^2}x

The Attempt at a Solution

The first thing I thought was using the general solution of x(t) = A cos(wt + psi), differentiating to find the velocity (after the phase angle, or is this even needed?), and then using the value in K.E = 1/2mv^2?

Please help!

 

[JaWiB]

That might work, but you also need to calculate the potential energy stored in the spring. You may already know the formula or you can find it by integrating the spring force over displacement (work done by the spring is minus the potential energy stored in the spring)

 

[CanIExplore]

The equation you've written

ma = -w^2x

should read a=-w²x right?

ma=-kx
a=-(k/m)x
a=-w²x
where w²=(k/m)

Following JaWiB, potential energy is equal to the negative line integral of the force. U=-$$\int F dl$$, which will come out to a familiar equation you've probably seen before. Since you're given the maximum displacement (amplitude), consider the energy of the particle at its maximum displacement and at the point of equilibrium (x=0). At what points is the energy of the particle purely kinetic or purely potential? Then consider conservation of energy.

 

[Redbelly98]

No integrals are needed here. The OP was right on track:

The Attempt at a Solution

The first thing I thought was using the general solution of x(t) = A cos(wt + psi), differentiating to find the velocity (after the phase angle, or is this even needed?), and then using the value in K.E = 1/2mv^2?

Don't worry about the phase angle, you may assume it's zero for simplicity.
The tricky part is what to do about the potential energy. At some point (value of x) the potential energy is zero, which simplifies the calculation of total energy.

 

[rdl]

I would approach this problem by first understanding the concept of simple harmonic motion and its equation of motion, which is given by ma = -w^2x. This equation tells us that the acceleration of the mass is directly proportional to its displacement from equilibrium, with a constant of proportionality being the square of the angular frequency (w^2).

Next, I would use the given amplitude and frequency to calculate the angular frequency (w) using the formula w = 2πf, where f is the frequency. In this case, w = 2π(16Hz) = 32π rad/s.

Then, I would use the general solution of simple harmonic motion, x(t) = A cos(wt + psi), to find the displacement of the mass at any given time. In this case, the amplitude (A) is given as 1cm, and the phase angle (psi) can be assumed to be zero since it is not specified.

Using this equation, we can find the velocity of the mass at any given time by differentiating x(t) with respect to time. This will give us the expression for velocity, v(t) = -Aw sin(wt + psi). Again, since psi is assumed to be zero, this simplifies to v(t) = -Aw sin(wt).

Finally, we can calculate the total energy of the mass by adding the kinetic energy (KE) and potential energy (PE) components. The kinetic energy can be calculated using the formula KE = 1/2mv^2, and the potential energy can be calculated using the formula PE = 1/2kx^2, where k is the spring constant.

In this case, since the mass is given as 2kg and the displacement varies with time, we can find the average values of velocity and displacement over one cycle and use them to calculate the total energy. This would give us:

KE = 1/2(2kg)(-Aw)^2 = 1/2(2kg)(32π)^2(1cm)^2 = 102.4π^2 mJ
PE = 1/2(2kg)(32π)^2(1cm)^2 = 102.4π^2 mJ

Therefore, the total energy of the mass in simple harmonic motion would be 204.8π^2 mJ.

In conclusion, by