Question

For a particle executing \(\mathrm{S} . \mathrm{H} \mathrm{M} .\), when the potential energy of the oscillator becomes \(1 / 8\) the maximum potential energy, the displacement of the oscillator in terms of amplitude A will be........... (A) \((\mathrm{A} / \sqrt{2})\) (B) \(\\{\mathrm{A} /(2 \sqrt{2})\\}\) (C) \((\mathrm{A} / 2)\) (D) \(\\{\mathrm{A} /(3 \sqrt{2})\\}\)

Short answer

The displacement of the oscillator in terms of amplitude A will be \((A / \sqrt{2})\).

Step-by-step solution

Relationship between potential energy and displacement

In simple harmonic motion (SHM), potential energy (U) is given by the formula: \[U = \frac{1}{2} kx^2\] where k is the spring constant, and x is the displacement from the equilibrium position. Also, the maximum potential energy (U_max) occurs when the displacement is at its maximum, which is equal to the amplitude (A), so: \[U_{\text{max}} = \frac{1}{2} kA^2\]

Determine the potential energy given in the exercise

We are given that the potential energy becomes 1/8 of the maximum potential energy. So, we have: \[U = \frac{1}{8} U_{\text{max}}\]

Substitute potential energy expressions and solve for displacement

Now, we substitute the expressions for U and U_max from Steps 1 and 2: \[\frac{1}{2} kx^2 = \frac{1}{8} \cdot \frac{1}{2} kA^2\] We can now cancel out the k and some constants: \[x^2 = \frac{1}{4} A^2\] Taking the square root of both sides, we obtain the displacement x in terms of amplitude A: \[x = \frac{A}{\sqrt{2}}\] So the correct answer is (A) \((A / \sqrt{2})\).

Key concepts

Potential Energy

Potential energy in the context of simple harmonic motion (SHM) is a critical concept that reflects the energy stored in the system due to the object's position. In SHM, such as with a spring-mass system, potential energy is defined by the formula \( U = \frac{1}{2} kx^2 \), where \( k \) is the spring constant and \( x \) is the displacement. This formula indicates that potential energy depends quadratically on the displacement. It grows as the displacement increases and is zero at the equilibrium point. Understanding this relationship helps us grasp how energy oscillates between potential and kinetic forms in SHM, contributing to the motion's periodic nature.
In the exercise, we look at scenarios where potential energy is a fraction of its maximum. This insight allowed us to determine how displacement correlates with potential energy, illustrating energy conservation principles in SHM.

Amplitude

Amplitude is a fundamental term in SHM, representing the maximum extent of oscillation from the system's equilibrium position. In essence, it is the furthest point the oscillating particle reaches before reversing direction.
In the exercise, amplitude (A) is used to define the maximum potential energy, \( U_{\text{max}} = \frac{1}{2} kA^2 \). This illustrates how amplitude not only dictates the string's maximum stretch but also determines the upper limit of the potential energy. Knowing the amplitude, you can predict several characteristics of the motion, such as potential and kinetic energy exchanges and the maximum speed.

Displacement

Displacement in the realm of SHM refers to how far an oscillating object is from its mean (equilibrium) position at any point in time. It is precisely this displacement that creates potential energy within the system.
In our problem, the relationship \( U = \frac{1}{2} kx^2 \) helps establish how displacement influences potential energy. The task asked for the displacement when potential energy is one-eighth of its maximum value, requiring manipulating this relationship. Solving \( x = \frac{A}{\sqrt{2}} \) demonstrates how displacement is sometimes expressed as a fraction or function of amplitude, highlighting its importance in the energy dynamics of SHM.

Spring Constant

The spring constant, denoted by \( k \), is a measure of a spring's stiffness in SHM. It plays a pivotal role by linking force, displacement, and energy in the examined spring system.
The formula for potential energy \( U = \frac{1}{2} kx^2 \) illustrates this connection, showing how the spring constant determines how much energy is stored in the spring as it is compressed or stretched. A higher spring constant means a stiffer spring that accumulates more potential energy for a given displacement.

• The spring constant is unique to each spring, determined by material and dimensions.
• It correlates directly with the restoring force the spring exerts; a greater \( k \) results in a stronger restoring force.

This helps us understand why in SHM, the interplay between spring constant and displacement dictates the energy transformations that characterize the motion.