91Ó°ÊÓ

In Simple Harmonic Motion (SHM), displacement refers to how far an object is from its equilibrium position at any given time. This equilibrium position is the center of the motion where the force is zero. The displacement in SHM can be expressed as a sine or cosine function, depending on the starting position and time. The maximum displacement from equilibrium is known as the amplitude, denoted as \(A\). When solving problems related to SHM, understanding displacement is crucial because it impacts both kinetic and potential energy.

Kinetic energy in SHM is linked to how fast the object is moving. The faster the object moves, the more kinetic energy it possesses. The formula for kinetic energy \(K\) in SHM is:
\[ K = E - U \] where \(E\) is the total energy, and \(U\) is the potential energy. In the given exercise, the displacement is half the amplitude, \(x = \frac{A}{2}\). At this displacement:
\[ U = \frac{1}{2} k \left(\frac{A}{2}\right)^2 = \frac{1}{8} k A^2 \]
Using the expression for total energy, \(E = \frac{1}{2} k A^2\), the fraction of kinetic energy at this displacement is:
\[ \frac{K}{E} = 1 - \frac{U}{E} = 1 - \frac{1}{4} = \frac{3}{4} \]
Hence, at half the amplitude, three-fourths of the total energy is kinetic.

Potential energy in SHM is associated with the object’s position relative to the equilibrium. When it's displaced, it stores potential energy based on how compressed or stretched the spring (or analogous system) is. The formula for potential energy \(U\) is:
\[ U = \frac{1}{2} k x^2 \]
For the displacement \( x = \frac{A}{2} \), the potential energy fraction is calculated as:
\[ U = \frac{1}{2} k \left(\frac{A}{2}\right)^2 = \frac{1}{8} k A^2 \]
To find the fraction of total energy that is potential energy, use:
\[ \frac{U}{E} = \frac{\frac{1}{8} k A^2}{\frac{1}{2} k A^2} = \frac{1}{4} \]
Therefore, at half the amplitude, one-fourth of the total energy is potential.

Energy in SHM can be equally divided between kinetic and potential energy at a specific displacement. To find this point, we set the kinetic energy \(K\) equal to the potential energy \(U\). Given the total energy \(E\) is the sum of kinetic and potential energy, each form of energy will be half the total energy:
\[ U = K = \frac{E}{2} \]
Substituting \( \frac{E}{2} \) for potential energy in the formula:
\[ \frac{1}{2} k x^2 = \frac{1}{2} \left( \frac{1}{2} k A^2 \right) \]
Simplifying, we get:
\[ x^2 = \frac{A^2}{2} \]
Thus, the displacement for equal energy distribution is:
\[ x = \frac{A}{\sqrt{2}} \]
This value indicates the displacement at which the system's mechanical energy is equally shared between kinetic and potential forms.