91Ó°ÊÓ

Conservation of momentum is a core principle in physics that describes how the total momentum of a closed system remains constant unless acted on by an external force. In the given problem, initially, the only moving object is the bullet. Thus, the entire system's momentum is conserved just before and immediately after the bullet embeds into the block.

To understand this better:

• Before impact: Only the bullet is moving, so the total momentum is the momentum of the bullet, calculated as mass (\( m_b = 0.0080 \, \text{kg} \)) times velocity (\( v_b = 280 \, \text{m/s} \)).
• After impact: The bullet sticks to the block, moving together. So, their combined mass is \( m_b + M = 1.000 \, \text{kg} \), and their shared velocity is \( v_f \).

Applying the conservation of momentum principle:\[m_b v_b = (m_b + M) v_f \]This equation allows the calculation of the final velocity (\( v_f \)). This principle ensures the total momentum of the system stays the same before and after the collision.

The spring constant, denoted by \( k \), is a measure of a spring's stiffness. In the given exercise, after the bullet becomes embedded in the block, the kinetic energy of the combined system is converted into potential energy when the spring is compressed. To find the spring constant, we equate the initial kinetic energy of the moving block and bullet with the potential energy stored in the spring.

Here's how:

• Kinetic energy just after collision is given by: \( \frac{1}{2}(m_b + M)v_f^2 \)
• At maximum spring compression, the kinetic energy becomes potential energy: \( \frac{1}{2}kx^2 \)
• By equating these two energies: \( \frac{1}{2}(m_b + M)v_f^2 = \frac{1}{2}kx^2 \)

Solving for \( k \), the following formula provides its value:\[k = \frac{(m_b + M) v_f^2}{x^2} = \frac{1.000 \times 2.24^2}{0.15^2} \approx 223.78 \, \text{N/m}\]This result tells us how rigid or supportive the spring is during compression, crucial for calculating other aspects of the system like the period of motion.

The transformation between kinetic and potential energy is a vital concept in understanding simple harmonic motion (SHM). Initially, the bullet's kinetic energy is transferred to the block, setting the system in motion. This kinetic energy is gradually converted into potential energy at the peak spring compression, allowing us to analyze the SHM.

Think of it this way:

• After the bullet fuses with the block, they share a kinetic energy governed by the equation \( \frac{1}{2}(m_b + M)v_f^2 \).
• As the system compresses the spring to its maximum, this energy converts to potential: \( \frac{1}{2}kx^2 \).
• Once released, the potential energy changes back to kinetic energy, enabling the back-and-forth motion of SHM.

Understanding the exchange of energy forms between kinetic and potential during SHM enables predictions of the system’s dynamics, exemplified by the calculation of the motion's period using:\[T = 2\pi \sqrt{\frac{m_b + M}{k}} \approx 0.420 \, \text{s}\]An insight into these energy transformations enhances our grasp of the rhythmic oscillation characteristic of systems in SHM.