A block of mass m moving at a speedb v compresses a spring through a distance x before is halved.Find the spring constant of the spring.

To find the spring constant of the spring when a block of mass \( m \) compresses it through a distance \( x \) before coming to a stop, we can utilize the principles of energy conservation. The kinetic energy of the block is converted into the potential energy stored in the spring during the compression. Let's break this down step by step.

Understanding the Energy Transformation

Initially, the block has kinetic energy due to its motion, which can be expressed as:

• Kinetic Energy (KE) = \( \frac{1}{2} mv^2 \)

When the block compresses the spring by a distance \( x \), it comes to a stop, and all of its kinetic energy is converted into the potential energy stored in the spring. The potential energy (PE) in a compressed spring is given by:

• Potential Energy (PE) = \( \frac{1}{2} k x^2 \)

Setting Up the Equation

According to the law of conservation of energy, the kinetic energy of the block before it compresses the spring is equal to the potential energy stored in the spring at maximum compression:

\( \frac{1}{2} mv^2 = \frac{1}{2} k x^2 \)

Solving for the Spring Constant

To find the spring constant \( k \), we can rearrange the equation. First, we can eliminate the \( \frac{1}{2} \) from both sides:

\( mv^2 = k x^2 \)

Next, we isolate \( k \):

\( k = \frac{mv^2}{x^2} \)

Final Expression for the Spring Constant

Thus, the spring constant \( k \) can be expressed as:

Spring Constant (k) = \( \frac{mv^2}{x^2} \)

Example Calculation

Let’s say we have a block with a mass of 2 kg moving at a speed of 4 m/s, and it compresses the spring by 0.5 m. We can plug these values into our formula:

• Mass \( m = 2 \, \text{kg} \)
• Speed \( v = 4 \, \text{m/s} \)
• Compression \( x = 0.5 \, \text{m} \)

Now, substituting these values into the equation:

\( k = \frac{2 \times (4)^2}{(0.5)^2} = \frac{2 \times 16}{0.25} = \frac{32}{0.25} = 128 \, \text{N/m} \)

So, the spring constant \( k \) would be 128 N/m. This means that for every meter the spring is compressed, it exerts a force of 128 Newtons in the opposite direction.

In summary, by applying the conservation of energy principle, we can derive the spring constant from the mass and speed of the block as it compresses the spring. This relationship is fundamental in understanding how energy is transferred and stored in mechanical systems.