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In simple harmonic motion (SHM), mechanical energy plays a crucial role. It represents the total energy of the system, combining both kinetic and potential energies. Mechanical energy remains constant as the toy oscillates back and forth on the spring. This constancy is due to the ideal nature of SHM, where no energy is lost to friction or air resistance.

The mechanical energy in SHM, denoted as \(E\), is given by:

• \(E = K + U\)

where \(K\) is the kinetic energy and \(U\) is the potential energy. This equation tells us that as the toy moves, energy transforms between kinetic and potential forms, but the total stays the same. This principle is essential for understanding how the system behaves dynamically during motion.

The spring constant, often represented by \(k\), is a measure of a spring's stiffness. In the context of SHM, it's critical because it determines how much force is required to stretch or compress the spring a certain distance. The spring constant is expressed in units of newtons per meter (N/m).

For our exercise, the spring constant is given as 300 N/m. This high value indicates a relatively stiff spring that exerts considerable force, making it responsive to small displacements. Knowing the spring constant is vital for calculating both potential energy and the amplitude of motion. It's also used in the formula:

• \(U = \frac{1}{2} k x^2\)

where \(x\) is the displacement from the equilibrium position.

The amplitude of motion, noted as \(A\), is the maximum displacement from the equilibrium position in simple harmonic motion. Amplitude is integral to describing the extent of oscillation. In our problem, we calculate the amplitude using the total mechanical energy, because it's when the spring is fully compressed or extended, and the system has all its energy as potential.

The formula linking energy and amplitude is:

• \(E = \frac{1}{2} k A^2\)

Solving for \(A\) helps us understand the toy's motion range, showing how far it moves from the center point on each side during its oscillation.

In simple harmonic motion, the maximum speed occurs when the object passes through the equilibrium position. At this point, all the system's mechanical energy is kinetic, with potential energy reaching zero.

The calculation of maximum speed, \(v_{max}\), involves rearranging the mechanical energy equation:

• \(E = \frac{1}{2} m v_{max}^2\)

By substituting known values, maximum speed is determined, which helps us gauge how fast the toy moves as it crosses its equilibrium position. Understanding \(v_{max}\) is essential for predicting dynamic behavior in SHM.

Kinetic energy in the context of SHM refers to the energy associated with an object’s motion. It depends on the mass of the object and its velocity. In our scenario, the kinetic energy changes as the toy moves along the spring.

The formula for kinetic energy \(K\) is:

• \(K = \frac{1}{2} m v^2\)

As the toy speeds up and slows down through its path, kinetic energy increases when moving towards the equilibrium position and decreases when moving away. Everything converts between kinetic and potential energy while maintaining the total energy constant.

Potential energy in simple harmonic motion is stored as the spring is compressed or stretched. It represents the energy stored due to the position of the toy relative to its equilibrium point. This energy is pivotal to the cyclic nature of SHM.

The formula for potential energy \(U\) is:

• \(U = \frac{1}{2} k x^2\)

where \(x\) is the displacement from the equilibrium position. As the toy moves, potential energy reaches its peak when the spring is at either its most compressed or most extended state. Potential energy is minimal when the toy passes through equilibrium, converting to kinetic energy here.