91Ó°ÊÓ

In Simple Harmonic Motion (SHM), an object like our glider reaches its maximum speed when it is passing through the equilibrium position. This is because, at this point, the potential energy is completely converted into kinetic energy. The formula to determine this maximum speed, often denoted as \( v_{max} \), is given by:

• \( v_{max} = A \omega \)

where \( A \) is the amplitude of motion, and \( \omega \) represents the angular frequency.

The angular frequency \( \omega \) can be calculated using the formula:

• \( \omega = \sqrt{\frac{k}{m}} \)

where \( k \) is the spring constant, and \( m \) is the mass of the glider.

From the given values, \( A = 0.040 \, \text{m} \) and \( \omega = 30 \, \text{rad/s} \), we find that the maximum speed \( v_{max} = 1.2 \, \text{m/s} \). This maximum speed is the highest speed the glider reaches as it oscillates back and forth.

Angular frequency is a key concept in SHM, determining how fast the oscillations occur. It is denoted by \( \omega \) and is measured in radians per second (rad/s). Angular frequency is related to the spring constant \( k \) and the mass \( m \) by the formula:

• \( \omega = \sqrt{\frac{k}{m}} \)

This equation shows the intrinsic connection between the stiffness of the spring and the inertia of the mass. For our glider, with \( k = 450 \, \text{N/m} \) and \( m = 0.500 \, \text{kg} \), the angular frequency \( \omega \) is calculated as \( 30 \, \text{rad/s} \).

This value is important because it influences both the maximum speed and the maximum acceleration of the glider during its oscillation.

In SHM, the mechanical energy remains constant throughout the motion, consisting of both kinetic and potential energy. The total mechanical energy \( E \) can be expressed as:

• \( E = \frac{1}{2} k A^2 \)

where \( k \) is the spring constant and \( A \) is the amplitude.

This energy expression highlights that the energy is proportional to both the square of the amplitude and the stiffness of the spring. For the glider with \( A = 0.040 \, \text{m} \) and \( k = 450 \, \text{N/m} \), we find the total mechanical energy is \( 0.36 \, \text{J} \). This energy value is maintained throughout the oscillation, ensuring that the system is conservative with interchangeable kinetic and potential energy.

Acceleration in Simple Harmonic Motion is tied to the position of the oscillating object and the spring constant. The maximum acceleration is experienced at the extreme positions of motion, calculated using:

• \( a_{max} = A \omega^2 \)

For the given values, \( a_{max} = 36 \, \text{m/s}^2 \), demonstrating how the acceleration peaks when the glider is furthest from the equilibrium point.

Moreover, the acceleration at any position \( x \) can be computed as:

• \( a = -\frac{k}{m}x \)

At \( x = -0.015 \, \text{m} \), the acceleration becomes \( 13.5 \, \text{m/s}^2 \). These calculations show that acceleration is greatest at the turning points and changes direction as the glider moves through the equilibrium point, following Hooke's Law principles.