91Ó°ÊÓ

Gravitational force is an essential concept when studying work and energy, particularly in the context we have here. It is the force of attraction between a mass and the Earth, acting downward towards the center of the Earth. The work done by gravitational force, therefore, can be calculated using the formula: \[ W_g = m imes g imes h \]

• \( m \): the mass of the object, measured in kilograms (kg).
• \( g \): the acceleration due to gravity, approximately \(9.81 \,\text{m/s}^2\) on Earth's surface.
• \( h \): the vertical displacement or the height from which the object falls, in meters (m).

In our exercise, we find that the block has a mass of \(0.25 \, \text{kg}\) and falls a height of \(0.12 \, \text{m}\). Substituting these values into our formula, the gravitational work is found to be \(0.294 \, \text{J}\). This calculation shows how gravitational energy is converted into mechanical work, providing a fundamental understanding necessary for analyzing work and energy in physics.

The spring force operates under Hooke's Law, which states that the force exerted by a spring is directly proportional to its compression or stretch. The equation for the force by the spring is given by: \[ F_s = -k imes x \]

• \( k \): the spring constant, denoting the stiffness of the spring, typically in \(\text{N/m}\).
• \( x \): the compression or elongation of the spring from its natural length, in meters.

The work done by the spring force can be calculated using the formula: \[ W_s = -\frac{1}{2} k x^2 \] Here, the negative sign indicates that the work done by the spring force is against the direction of displacement of the block. In the context of our exercise, the spring constant is \(250 \, \text{N/m}\), and the spring compresses by \(0.12 \, \text{m}\). This calculation results in work done by the spring force of \(-1.8 \, \text{J}\). The energy stored is potential energy within the spring, playing a key role in the energy conservation of mechanical systems.

Energy conservation is a fundamental principle stating that energy cannot be created or destroyed, only transformed from one form to another. In the case of the block and spring system, the energy transforms from gravitational potential energy to kinetic energy, and finally to spring potential energy. The principle of mechanical energy conservation is stated as: \[ mgh = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 \]

• Initial Gravitational Energy (\(mgh\)): Energy gained from height.
• Kinetic Energy (\(\frac{1}{2}mv^2\)): Energy due to motion just before impact.
• Spring Potential Energy (\(\frac{1}{2}kx^2\)): Energy stored in the spring at maximum compression.

Using these energies, we found the speed of the block just before the spring compression using the energy conservation principle, calculated as \(1.96 \, \text{m/s}\). This approach demonstrates how energy transformation within a closed system maintains total energy.

The compression of a spring is an important measure of how much energy is stored within it. In our exercise, after accounting for an increased initial speed, the behavior of the block and spring system changes due to energy conservation. Initially, the block had a certain potential and kinetic energy, which, upon impact, interacted with the spring's resistance and caused compression. With the impact speed doubled to \(3.92 \, \text{m/s}\), we revisit our energy equations: \[ \frac{1}{2}mv_i^2 = mgh + \frac{1}{2}kx^2 \] Solving the quadratic equation formed gives a new maximum compression for the spring. This demonstrates how doubling the kinetic energy of the block alters the mechanics of the system, resulting in a larger compression of approximately \(0.24 \, \text{m}\). Understanding the relationship between speed, energy, and compression helps students grasp the dynamic interactions within mechanical systems and the nature of spring-based energy storage.