Problem 26 - Physics

Question

A rifle fires a \(2.10 \times 10^{-2} \mathrm{~kg}\) pellet straight upward, because the pellet rests on a compressed spring that is released when the trigger is pulled. The spring has a negligible mass and is compressed by \(9.10 \times 10^{-2} \mathrm{~m}\) from its unstrained length. The pellet rises to a maximum height of \(6.10 \mathrm{~m}\) above its position on the compressed spring. Ignoring air resistance, determine the spring constant.

Step-by-Step Solution

Answer

The spring constant is approximately 304 N/m.

Step 1: Identify and List Known Quantities

First, note the given quantities: - Mass of the pellet, \( m = 2.10 \times 10^{-2} \text{ kg} \)- Compression distance of the spring, \( x = 9.10 \times 10^{-2} \text{ m} \)- Maximum height reached by the pellet, \( h = 6.10 \text{ m} \).

Step 2: Apply Conservation of Energy Principle

Use the conservation of mechanical energy where the potential energy stored in the spring is converted to gravitational potential energy at maximum height. Express the equation:\[ \frac{1}{2} k x^2 = mgh \]where \( k \) is the spring constant and \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity.

Step 3: Solve for the Spring Constant

Rearrange the equation from the previous step to solve for the spring constant \( k \): \[ k = \frac{2mgh}{x^2} \]Substitute the values to find \( k \): \[ k = \frac{2 \cdot (2.10 \times 10^{-2}) \cdot 9.81 \cdot 6.10}{(9.10 \times 10^{-2})^2} \]

Step 4: Perform the Calculation

Calculate \( k \) using the provided values:\[ k = \frac{2 \cdot 2.10 \times 10^{-2} \cdot 9.81 \cdot 6.10}{(9.10 \times 10^{-2})^2} = \frac{2.52}{0.008281} \approx 304 \text{ N/m} \].

Key Concepts

Spring Constant

The spring constant is a crucial part of understanding how springs behave in physics. It tells us how stiff or flexible a spring is. The spring constant, often denoted as \( k \), is measured in newtons per meter (N/m). It reflects the force needed to compress or extend the spring by one meter.

In the context of the exercise, we are dealing with a compressed spring that launches a pellet upward. The compression of the spring stores potential energy, which is then converted into kinetic energy as the spring is released, propelling the pellet into the air.

To find the spring constant, we use the energy conservation principle that relates the stored potential energy in the spring to the gravitational potential energy at the highest point the pellet reaches. The formula used is:

• Stored Energy in the Spring: \( \frac{1}{2} k x^2 \)
• Gravitational Potential Energy: \( mgh \)

By finding \( k \), we reveal how efficiently the spring can perform its intended function: converting its stored energy into motion.

Gravitational Potential Energy

Gravitational potential energy is the energy that an object possesses because of its position in a gravitational field. It's like lifting a ball; higher positions mean more energy stored. The formula for gravitational potential energy is \( mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity (9.81 m/s² on Earth), and \( h \) is the height.

In our exercise, the pellet reaches a height of 6.10 meters after being launched from the spring. This height represents the maximum potential energy the pellet achieves thanks to the spring's stored power being transferred to it.

As the pellet gains height, it trades kinetic energy for gravitational potential energy until it reaches its peak. https://ru.bookmate.com/books/TkQ2BVHJ

• This energy can be calculated using the known mass of the pellet, gravity, and the achieved height.

Understanding this exchange of energy forms is crucial for knowing how energy conserves and transforms in physical systems.

Mechanical Energy

Mechanical energy is the sum of potential and kinetic energy in a system, one of the basic principles in physics. It includes various forms of energy in motion and stored energy ready to be used. The conservation of mechanical energy is key in systems without external forces like friction or air resistance.

For the exercise, mechanical energy conservation means that all the energy the spring stored must be transferred to the pellet without loss. This spring-generated energy propels the pellet to its peak. We see this as:

• The initial energy in the spring, given by \( \frac{1}{2} k x^2 \),
• becomes the potential energy at the pellet's highest point, \( mgh \).

In ideal scenarios, characterized by perfect conservation, the total mechanical energy remains the same before and after events like shooting the pellet. It provides a way to calculate unknowns like the spring constant or the potential position energy just by considering how energy shifts and conserves in the system.