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Chapter 10: Problem 32

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.

Short Answer

Expert verified
The spring constant \(k\) is approximately \(303.48 \text{ N/m}\).

Step by step solution

01

Understanding the Problem

We are given a pellet with mass \(2.10 \times 10^{-2} \text{ kg}\), a spring compressed by \(9.10 \times 10^{-2} \text{ m}\), and a maximum height reached by the pellet of \(6.10 \text{ m}\). We are asked to find the spring constant \(k\).
02

Identify the Energy Transformation

The mechanical energy transformation in this problem is from elastic potential energy in the spring to gravitational potential energy when the pellet reaches its maximum height. The equations involved are the elastic potential energy \(E_{spring} = \frac{1}{2} k x^2\) and gravitational potential energy \(E_{gravity} = mgh\).
03

Write the Energy Conservation Equation

At the lowest point, all energy is stored as elastic potential energy in the spring. At the maximum height, all energy is gravitational. Using the conservation of energy: \[ \frac{1}{2} k x^2 = mgh \] where \(x = 9.10 \times 10^{-2} \text{ m}\), \(m = 2.10 \times 10^{-2} \text{ kg}\), \(g = 9.81 \text{ m/s}^2\), and \(h = 6.10 \text{ m}\).
04

Substitute Known Values

Substitute the known values into the energy conservation equation:\[ \frac{1}{2} k (9.10 \times 10^{-2})^2 = (2.10 \times 10^{-2})(9.81)(6.10) \]
05

Solve for the Spring Constant k

Rearrange the equation to solve for \(k\):\[ k = \frac{2 imes (2.10 \times 10^{-2})(9.81)(6.10)}{ (9.10 \times 10^{-2})^2} \]Calculate the value of \(k\).
06

Calculate and Find k's Value

Calculate the right side of the equation:\[ (2.10 \times 10^{-2} \times 9.81 \times 6.10) = 1.254582 \]\[((9.10 \times 10^{-2})^2 = 0.008281\]\[ k = \frac{2 \times 1.254582}{0.008281} = 303.48 \text{ N/m}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Energy Conservation
Energy conservation is a central concept in physics, focusing on how energy is neither created nor destroyed but rather transformed from one form to another. In our context, the energy transformation involves a spring and a pellet.
  • Initially, the energy is stored in the spring, compressed and ready to push the pellet upward.
  • As the spring releases, it converts its stored elastic potential energy into gravitational potential energy as the pellet rises.
Understanding this principle helps us solve many physics problems where energy shifts from one type to another while keeping the total energy constant in an isolated system.
Elastic Potential Energy
Elastic potential energy is the energy stored in elastic materials, like springs, when they are stretched or compressed.
  • For springs, this energy can be calculated using the formula: \(E_{spring} = \frac{1}{2} k x^2\).
  • Here, \(k\) is the spring constant, measuring the stiffness of the spring, and \(x\) is the compression length.
This form of energy is crucial in launching the pellet upward, and understanding it allows us to calculate how far or how high an object will move when released.
Gravitational Potential Energy
Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is calculated using the formula: \(E_{gravity} = mgh\).
  • \(m\) represents the mass of the object, \(g\) is the acceleration due to gravity (approximately \(9.81 \text{ m/s}^2\) on Earth), and \(h\) is the height from a reference point.
  • This form of energy reaches its maximum when the object, like the pellet, is at its highest point.
Understanding gravitational potential energy is essential to determine how high the pellet will rise using the energy derived from the spring.
Physics Problem Solving
Physics problem solving often involves breaking down complex problems into manageable parts, as shown in our spring constant calculation example.
  • First, identify the known quantities and what needs to be found, like mass, height, and spring compression.
  • Use physics concepts such as energy conservation to set up equations that represent these transformations.
  • Substitute known values into the equations to find unknown variables, like the spring constant \(k\).
This structured approach helps solve various physics challenges, ensuring that each step logically follows from the previous one.

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Most popular questions from this chapter

Multiple-Concept Example 6 presents a model for solving this problem. As far as vertical oscillations are concerned, a certain automobile can be considered to be mounted on four identical springs, each having a spring constant of \(1.30 \times 10^{5} \mathrm{N} / \mathrm{m} .\) Four identical passengers sit down inside the car, and it is set into a vertical oscillation that has a period of 0.370 s. If the mass of the empty car is 1560 kg, determine the mass of each passenger. Assume that the mass of the car and its passengers is distributed evenly over the springs.

A spring is resting vertically on a table. A small box is dropped onto the top of the spring and compresses it. Suppose the spring has a spring constant of 450 N/m and the box has a mass of 1.5 kg. The speed of the box just before it makes contact with the spring is 0.49 m/s. (a) Determine the magnitude of the spring鈥檚 displacement at an instant when the acceleration of the box is zero. (b) What is the magnitude of the spring鈥檚 displacement when the spring is fully compressed?

A 1.0 \times 10 \(^{-3}\) -kg spider is hanging vertically by a thread that has a Young's modulus of \(4.5 \times 10^{9} \mathrm{N} / \mathrm{m}^{2}\) and a radius of \(13 \times 10^{-6} \mathrm{m} .\) Suppose that a \(95-\mathrm{kg}\) person is hanging vertically on an aluminum wire. What is the radius of the wire that would exhibit the same strain as the spider's thread, when the thread is stressed by the full weight of the spider?

A die is designed to punch holes with a radius of \(1.00 \times 10^{-2} \mathrm{m}\) in a metal sheet that is \(3.0 \times 10^{-3} \mathrm{m}\) thick, as the drawing illustrates. To punch through the sheet, the die must exert a shearing stress of \(3.5 \times 10^{8} \mathrm{Pa} .\) What force \(\overrightarrow{\mathbf{F}}\) must be applied to the die?

A 1.1-kg object is suspended from a vertical spring whose spring constant is \(120 \mathrm{N} / \mathrm{m}\). (a) Find the amount by which the spring is stretched from its unstrained length. (b) The object is pulled straight down by an additional distance of \(0.20 \mathrm{m}\) and released from rest. Find the speed with which the object passes through its original position on the way up.
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