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Frequency in simple harmonic motion is a vital parameter that tells us how many oscillations occur per unit of time. For a block sliding on a frictionless surface attached to a spring, the frequency can be determined from the spring constant and the mass of the block.
To calculate the frequency (\( f \)), we use the formula:\[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}.\]This equation indicates that frequency is directly influenced by the spring constant (\( k \)) and inversely by the mass (\( m \)). For our block with a given \( k = 480 \text{ N/m} \) and \( m = 1.2 \text{ kg} \), substituting the values gives us:\[ f = \frac{1}{2\pi} \sqrt{\frac{480}{1.2}} = \frac{10}{\pi} \approx 3.18 \text{ Hz}.\]
This means the block oscillates approximately 3.18 times per second.

Amplitude in simple harmonic motion represents the maximum displacement from the equilibrium position. To find the amplitude (\( A \)), we rely on the principle of energy conservation. Initially, at the equilibrium position, all the energy is kinetic, given by:\[ E_{\text{kinetic}} = \frac{1}{2} mv^2.\]At the point of maximum displacement, the energy is entirely potential:\[ E_{\text{potential}} = \frac{1}{2} k A^2.\]By equating kinetic and potential energy and solving for \( A \), we find:\[ A = \sqrt{\frac{mv^2}{k}} = \sqrt{\frac{1.2 \times (5.2)^2}{480}} \approx 0.26 \text{ m}.\]
This calculation shows that the block reaches a maximum displacement of about 0.26 meters from the central position during its motion.

The concept of energy conservation is fundamental in analyzing simple harmonic motion. It implies that the total mechanical energy in the system remains constant, comprising both kinetic and potential energies. As the block moves back and forth, the energy continually exchanges between kinetic and potential forms.

• At the equilibrium point (\( x = 0 \)), the energy is purely kinetic: \( E_{\text{kinetic}} = \frac{1}{2} mv^2 \).
• At maximum displacement (\( x = A \)), the energy converts entirely to potential form: \( E_{\text{potential}} = \frac{1}{2} k A^2 \).

The reliability of energy conservation allows us to derive other properties, such as amplitude, by using the equality of kinetic and potential energies at different phases of motion.

Angular frequency (\( \omega \)) is another crucial parameter in simple harmonic motion. It showcases how fast the oscillating object moves through its cycle in terms of radians per second. Derived from the spring constant and mass, it is mathematically represented as:\[ \omega = \sqrt{\frac{k}{m}}.\]For our given problem, calculating \( \omega \) provides:\[ \omega = \sqrt{\frac{480}{1.2}} = 20 \text{ rad/s}.\]
Moreover, angular frequency relates to the frequency (\( f \)) via the relation \( \omega = 2\pi f \).This means that for this system, not only does \( \omega \) define the speed of oscillation in radians, but it also ties into the periodic nature that \( f \) represents, enhancing our understanding of the motion's rhythm.