Problem 56 - College Physics

Question

A \(1.5-\mathrm{kg}\) box that is sliding on a frictionless surface with a speed of \(12 \mathrm{~m} / \mathrm{s}\) approaches a horizontal spring. (See Fig. 5.19.) The spring has a spring constant of \(2000 \mathrm{~N} / \mathrm{m}\). If one end of the spring is fixed and the other end changes its position, (a) how far will the spring be compressed in stopping the box? (b) How far will the spring be compressed when the box's speed is reduced to half of its initial speed?

Step-by-Step Solution

Answer

(a) 0.328 m, (b) 0.164 m.

Step 1: Identify the Energy Types

The box moving on a frictionless surface initially has kinetic energy and as it compresses the spring, the energy is transformed into elastic potential energy in the spring. No energy is lost due to friction.

Step 2: Formulate the Energy Conservation Principle

The kinetic energy of the box before compressing the spring is given by\[KE = \frac{1}{2}mv^2\]and the potential energy stored in the spring when compressed is\[PE = \frac{1}{2}kx^2\]where \(m = 1.5 \text{ kg}\), \(v = 12 \text{ m/s}\), and \(k = 2000 \text{ N/m}\).

Step 3: Calculate the Maximum Compression for Part (a)

For the box to stop, all the kinetic energy converts into spring potential energy:\[\frac{1}{2}mv^2 = \frac{1}{2}kx^2\]Simplifying gives:\[x^2 = \frac{mv^2}{k}\]Substitute the known values:\[x^2 = \frac{1.5 \times 12^2}{2000} = 0.108\]\[x = \sqrt{0.108} \approx 0.328 \text{ m}\]

Step 4: Calculate the Compression for Part (b)

When the speed is reduced to half (6 m/s) the relationship is:\[\frac{1}{2}m\left(\frac{v}{2}\right)^2 = \frac{1}{2}kx^2\]Simplifying gives:\[x^2 = \frac{m\left(\frac{v}{2}\right)^2}{k}\]Substitute the values:\[x^2 = \frac{1.5 \times 6^2}{2000} = 0.027\]\[x = \sqrt{0.027} \approx 0.164 \text{ m}\]

Step 5: State Final Results

For part (a), the box compresses the spring by approximately \(0.328 \text{ meters}\). For part (b), when the speed is reduced to half, the compression is approximately \(0.164 \text{ meters}\).

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is dependent on both the mass and speed of the object according to the formula:\[KE = \frac{1}{2}mv^2\]where:

• \(m\) is the mass of the object.
• \(v\) is its velocity.

For example, when a 1.5-kg box slides on a frictionless surface at a speed of 12 m/s, its kinetic energy is calculated by substituting the values into the formula. This gives:\[KE = \frac{1}{2} \times 1.5 \,\text{kg} \times (12\,\text{m/s})^2 = 108\,\text{J}\]This implies that the box has 108 Joules of energy due to its motion. Kinetic energy is often converted into other forms of energy, such as in this exercise, where it is converted into elastic potential energy while the box compresses a spring.

Elastic Potential Energy

Elastic potential energy is stored in objects that can be stretched or compressed. For a spring, the potential energy stored is determined by the extent to which it is compressed or stretched and the spring constant.When an object like a box compresses a spring, it gradually converts its kinetic energy into elastic potential energy, according to the formula:\[PE = \frac{1}{2}kx^2\]where:

• \(k\) is the spring constant.
• \(x\) represents the displacement from its equilibrium position.

In this problem, the spring absorbs the kinetic energy of the box as it slows down, compressing further. For the box to stop, all its kinetic energy transitions into energy stored in the spring. The potential energy in the spring equals the initial kinetic energy of the box, allowing us to calculate the extent of compression using energy conservation principles. This conversion is key to understanding how energy transitions maintain the larger principle of energy conservation in physics.

Spring Constant

The spring constant, denoted by \(k\), is a measure of the stiffness of a spring. It describes how much force is needed to compress or stretch the spring by a unit length. The unit for the spring constant is Newton per meter (N/m).In our exercise, the spring constant is given as 2000 N/m. This means that to compress or stretch the spring by one meter, a force of 2000 Newtons is required. The value of the spring constant directly influences how the spring stores energy.During compression, the spring constant appears in the formula for elastic potential energy:\[PE = \frac{1}{2}kx^2\]A higher spring constant indicates a stiffer spring, meaning more energy will be stored for the same amount of compression compared to a spring with a lower constant. This property is crucial for calculating how far the spring compresses when energy is transferred to it, as observed in various practical and theoretical physics problems. Understanding the spring constant helps in designing systems in engineering where controlled energy storage is necessary.