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Angular frequency, denoted by \( \omega \), is an essential concept in simple harmonic motion. It describes how quickly the object oscillates back and forth. For a spring-block system, the angular frequency is determined by the equation: \[\omega = \sqrt{\frac{k}{m}} \]where \( k \) is the spring constant and \( m \) is the mass of the block.In this exercise, we're dealing with a spring constant \( k = 480 \,\mathrm{N/m} \) and a mass \( m = 1.2 \,\mathrm{kg} \). Plugging these values into the formula gives us:\[\omega = \sqrt{\frac{480}{1.2}} = \sqrt{400} = 20 \,\mathrm{rad/s}\]This result shows that the block completes its oscillation at a rate of 20 radians per second. Understanding angular frequency is crucial as it directly influences how the motion progresses over time.

The spring constant \( k \) is a measure of the stiffness of a spring. It tells us how much force is needed to stretch or compress the spring by a unit length. In the realm of simple harmonic motion, this constant plays a pivotal role in determining both the angular frequency and the amplitude of motion.Typically, a higher spring constant indicates a stiffer spring, resulting in a higher angular frequency for a given mass. This is because it requires more force to displace the spring, leading to faster oscillations.In our exercise, the spring constant is \( k = 480 \,\mathrm{N/m} \). This value signifies that a force of 480 Newtons is required to stretch or compress the spring by one meter.The stiffer the spring, the quicker the system will oscillate back to its equilibrium position. This constant is crucial for understanding how the spring influences the motion's behavior and energy characteristics.

Conservation of mechanical energy is a fundamental principle used to understand simple harmonic motion in spring systems. It states that the total mechanical energy in a closed system remains constant, provided no external forces like friction are present.In the context of a block-spring system, the mechanical energy is shared between potential energy in the compressed or stretched spring and kinetic energy of the moving block. Mathematically, it's expressed as:\[E = \frac{1}{2} m v_0^2 = \frac{1}{2} k A^2\]where \( E \) is the total mechanical energy, \( m \) is the mass, \( v_0 \) is the initial velocity, \( k \) is the spring constant, and \( A \) is the amplitude of motion.In the given problem, initially, when the block is moving with a speed of \( 5.2 \,\mathrm{m/s} \), all its energy is kinetic. As the block reaches its furthest displacement (amplitude), all this energy converts into potential energy stored in the spring. By equating the initial kinetic energy to the potential energy at maximum displacement, we find:\[\frac{1}{2} (1.2)(5.2)^2 = \frac{1}{2} (480) A^2 \]Solving this provides the amplitude \( A \approx 0.260 \,\mathrm{m} \). This principle highlights how energy transitions between forms during oscillations, explaining the sustained motion of the block.