Problem 26 - Physics

Question

A rifle fires a \(2.10 \times 10^{-2}\) 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 30.40 N/m.

Step 1: Analyze Given Data

We are provided with a pellet mass of \(m = 2.10 \times 10^{-2}\, \text{kg}\), maximum height \(h = 6.10\, \text{m}\), and initial compression of the spring \(x = 9.10 \times 10^{-2}\, \text{m}\). We need to find the spring constant \(k\).

Step 2: Apply Energy Conservation Principle

Since the spring is released and the pellet reaches a maximum height, we can relate the potential energy from the spring to the gravitational potential energy at maximum height. This gives us the equation: \[ \frac{1}{2} k x^2 = mgh \] Where \(g = 9.8 \, \text{m/s}^2\) is the acceleration due to gravity.

Step 3: Solve for Spring Constant

Rearrange the equation from Step 2 to solve for the spring constant \(k\): \[ k = \frac{2mgh}{x^2} \]

Step 4: Substitute Known Values

Substitute \(m = 2.10 \times 10^{-2} \, \mathrm{kg}\), \(g = 9.8 \, \mathrm{m/s^2}\), \(h = 6.10 \, \mathrm{m}\), and \(x = 9.10 \times 10^{-2} \, \mathrm{m}\) into the formula for \(k\):\[ k = \frac{2 \times 2.10 \times 10^{-2} \times 9.8 \times 6.10}{(9.10 \times 10^{-2})^2} \]

Step 5: Calculate the Spring Constant

Perform the calculations: \[ k = \frac{2 \times 2.10 \times 10^{-2} \times 9.8 \times 6.10}{8.281 \times 10^{-3}} = \frac{2.51676 \times 10^{-1}}{8.281 \times 10^{-3}} \approx 30.40 \, \text{N/m} \] The spring constant \(k\) is approximately \(30.40 \, \text{N/m}\).

Key Concepts

Energy Conservation

In physics, energy conservation is a fundamental concept. It states that energy cannot be created or destroyed, only converted from one form to another. When the trigger is pulled, the stored energy in the compressed spring is converted to kinetic energy and then to gravitational potential energy as the pellet rises. This transformation helps us understand how different forms of energy interact in closed systems. It's like a magical exchange where energy seems to "travel" between forms, while the total amount remains constant!

Gravitational Potential Energy

Gravitational potential energy is energy an object possesses because of its position in a gravitational field. For our rifle pellet, the energy is highest at its peak height of 6.1 meters. This energy depends on three factors: mass, gravity, and height.

• Mass ( m ) - The higher the mass, the more gravitational potential energy.
• Gravity ( g ) - Constant at g = 9.8 \, \text{m/s}^2 on Earth's surface.
• Height ( h ) - The higher the object, the more energy it has.

At maximum height, the kinetic energy is zero, and all the energy from the spring converts into gravitational potential energy: \[ mgh \].

Kinetic Energy

Kinetic energy is the energy of motion. When the spring is released, it pushes the pellet, converting potential energy stored in the spring into kinetic energy.

• Kinetic energy depends on mass and velocity: \( \frac{1}{2} mv^2 \).
• The moment the spring is fully uncompressed, it's when the pellet has maximum kinetic energy.
• As the pellet rises, this kinetic energy is gradually converted into gravitational potential energy.

This relationship between motion and energy transformation is a core piece of understanding movement dynamics.

Spring Potential Energy

The spring potential energy is stored in the spring when compressed. It's a form of stored energy, ready to be transformed.

• Spring energy is given by: \( \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the compression distance.
• Higher compression or a stiffer spring leads to more potential energy.
• When released, this energy is converted to kinetic and then gravitational potential energy.

Understanding this energy gives insight into how springs can store and release energy efficiently.