Question

Which of the equation given below represents a S.H.M.? (A) acceleration \(=-\mathrm{k}(\mathrm{x}+\mathrm{a})\) (B) acceleration \(=\mathrm{k}(\mathrm{x}+\mathrm{a})\) (C) acceleration \(=\mathrm{kx}\) (D) acceleration \(=-\mathrm{k}_{0} \mathrm{x}+\mathrm{k}_{1} \mathrm{x}^{2}\) \\{Here \(\mathrm{k}, \mathrm{k}_{0}\) and \(\mathrm{k}_{1}\) are force constants and units of \(\mathrm{x}\) and a is meter \(\\}\)

Short answer

The short answer is: Only equation (C), which is \( a = kx \), represents a Simple Harmonic Motion, as it satisfies the condition where the acceleration is proportional to the displacement and directed towards the mean position.

Step-by-step solution

Equation (A)

The acceleration in equation (A) is given by: \( a = -k(x+a) \) Expanding this equation, we get: \( a = -kx - ka \) This equation represents a motion with the acceleration being proportional to the displacement and has an additional constant term. The presence of this constant term, however, means that this equation does not satisfy the condition of S.H.M. Therefore, equation (A) does not represent a S.H.M. #Step 2: Check Equation (B)#

Equation (B)

The acceleration in equation (B) is given by: \( a = k(x+a) \) Expanding this equation, we get: \( a = kx + ka \) Similar to equation (A), the acceleration is proportional to the displacement, and it also has a constant term. Moreover, the acceleration acts in the same direction as the displacement, which does not satisfy the condition of S.H.M. Therefore, equation (B) does not represent a S.H.M. #Step 3: Check Equation (C)#

Equation (C)

The acceleration in equation (C) is given by: \( a = kx \) This equation represents a motion with acceleration directly proportional to the displacement and directed oppositely. Since it satisfies the condition of S.H.M, equation (C) represents a S.H.M. #Step 4: Check Equation (D)#

Equation (D)

The acceleration in equation (D) is given by: \( a = -k_{0}x + k_{1}x^{2} \) This equation represents a motion with acceleration that is not only proportional to the displacement but also depends on the square of the displacement. This additional dependency does not satisfy the condition of S.H.M. Therefore, equation (D) does not represent a S.H.M. #Conclusion#: Out of the given equations, only equation (C) represents a Simple Harmonic Motion, since the acceleration is proportional to the displacement and directed towards the mean position.

Key concepts

Understanding Acceleration in SHM

Acceleration is a crucial component in understanding Simple Harmonic Motion (SHM). In SHM, acceleration is always directed towards the mean position, which is why it is sometimes referred to as restoring acceleration.
For an object in SHM, acceleration is proportional and opposite to the displacement from its equilibrium position. Mathematically, this can be expressed as \[ a = -kx \] where - \(a\) is the acceleration, - \(k\) is the force constant, - \(x\) is the displacement.
In SHM, only equation (C) from the original exercise fits this model, with the negative sign indicating that the acceleration is directed opposite to the displacement. This characteristic ensures that as displacement increases in one direction, the force magnifies in the opposite direction to return the object to its equilibrium state.

Exploring Displacement in Simple Harmonic Motion

Displacement in SHM refers to how far the system deviates from its mean position. It is a pivotal factor in determining the system's acceleration and velocity. In SHM, the displacement oscillates in a sinusoidal pattern, indicating that it repeats itself over regular intervals.
The maximum displacement from the mean position is known as the amplitude of the motion. This amplitude, along with the force constant, influences how quickly the system oscillates back to its central position. As shown in the solution, equation (C) reflects that displacement impacts acceleration linearly, preserving the proportional relationship vital in SHM.
Moreover, the time taken for one complete oscillation, known as the period, is constant in ideal SHM. It depends on properties such as mass and force constant but not on the amplitude of displacement. This consistency is critical for applications where predictable and repetitive motion is necessary, such as clocks or musical instruments.

Role of Force Constants in SHM

Force constants, often denoted by \(k\), determine the stiffness and responsiveness of the system in SHM. A higher force constant indicates a stiffer system that returns to equilibrium more quickly. In a spring-mass system, for example, the spring constant represents the force constant.
In the equations evaluated in the exercise, the force constant is vital for identifying SHM. For instance, in equation (C), where acceleration is defined as \[ a = -kx \], \(k\) modulates how strongly the system attempts to restore itself to equilibrium with increased displacement.
Measurement of force constants provides valuable insights into material properties or the potential energy within a system. Knowing the force constant allows for precise predictions about the motion of a system under SHM conditions, which is why it is fundamental not just theoretically, but also in practical applications like designing suspension systems or tuning instruments.