Hi there,
I'm sorry to hear that you're stuck on this problem. Simple harmonic oscillator problems can be tricky, but with a little bit of guidance, I'm sure you'll be able to solve it.
First, let's review the basics of a simple harmonic oscillator. It is a system where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction of the displacement. This results in a back-and-forth motion around the equilibrium point.
Now, let's tackle part (a) of the problem. When the displacement is one half the amplitude, the system is at its maximum potential energy and minimum kinetic energy. This is because at this point, the displacement from equilibrium is at its maximum, so the restoring force is also at its maximum, resulting in the highest potential energy. On the other hand, since the velocity is zero at this point, the kinetic energy is also zero.
To calculate the specific values, we can use the equations for potential and kinetic energy in a simple harmonic oscillator:
Potential energy = (1/2)kx^2
Kinetic energy = (1/2)mv^2 = (1/2)m(ωA)^2sin^2(ωt)
Where k is the spring constant, x is the displacement, m is the mass, ω is the angular frequency, A is the amplitude, and t is time.
Since we know that the displacement is one half the amplitude, we can substitute x = 0.5A into the potential energy equation:
Potential energy = (1/2)k(0.5A)^2 = (1/8)kA^2
And since the kinetic energy is zero at this point, we can simply write:
Kinetic energy = 0
For part (b) of the problem, we need to find the displacement at which the kinetic energy is equal to the potential energy. To do this, we can set the equations for potential and kinetic energy equal to each other and solve for x:
(1/2)kx^2 = (1/2)m(ωA)^2sin^2(ωt)
Since we are looking for the point where the kinetic energy equals the potential energy, we can set the two equations equal to each other and solve for x:
(1/2)kx^2 = (1/8)kA^2
x^2 = (1/4
