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A block of mass 5.0 kg slides without friction at a speed of \(8.0 \mathrm{~m} / \mathrm{s}\) on a horizontal table surface until it strikes and sticks to a horizontal spring (with spring constant of \(k=2000 . \mathrm{N} / \mathrm{m}\) and very small mass \(),\) which in turn is attached to a wall. How far is the spring compressed before the mass comes to rest? a) \(0.40 \mathrm{~m}\) c) \(0.30 \mathrm{~m}\) e) \(0.67 \mathrm{~m}\) b) \(0.54 \mathrm{~m}\) d) \(0.020 \mathrm{~m}\)

Short Answer

Expert verified
a) 0.40 m b) 0.35 m c) 0.48 m d) 0.24 m Answer: a) 0.40 m

Step by step solution

01

Identify the initial and final energies of the system

Initially, the block has kinetic energy and the spring has no elastic potential energy, as it is not compressed. In the end, the block comes to rest, so it has no kinetic energy, while the spring has elastic potential energy due to compression.
02

Write down the expressions for kinetic energy and elastic potential energy

The kinetic energy of the block can be expressed as: \(KE_{initial} = \frac{1}{2}mv^2\) The elastic potential energy of the spring can be expressed as: \(PE_{spring} = \frac{1}{2}kx^2\) where \(m\) is the mass of the block, \(v\) is the initial velocity, \(k\) is the spring constant, and \(x\) is the compression of the spring.
03

Set up the law of conservation of energy equation

According to the law of conservation of energy, the total energy of the system remains constant. Therefore, we can write the equation as: \(KE_{initial} = PE_{spring}\)
04

Substitute the given values into the equation and solve for the compression

Now, we can plug in the given values for mass, initial velocity, and spring constant: \(\frac{1}{2}(5.0 \mathrm{kg})(8.0 \mathrm{ m/s})^2 = \frac{1}{2}(2000 \mathrm{ N/m})x^2\) Solve this equation to find the compression of the spring: \((5.0)(8.0)^2 = (2000)x^2\)
05

Calculate the compression of the spring

Now, we can solve for \(x\): \(x^2 = \frac{(5.0)(8.0)^2}{2000} = 0.16\) \(x = \sqrt{0.16} = 0.4 \mathrm{m}\) The spring is compressed by 0.4 meters before the mass comes to rest. The correct answer is a) \(0.40 \mathrm{m}\).

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Key Concepts

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

Kinetic Energy
Kinetic energy is the energy possessed by an object due to its motion. It depends on two main factors: the mass of the moving object and its velocity. Mathematically, kinetic energy (\(KE\text{)\text{)}} can be expressed as \(KE = \frac{1}{2}mv^2\), where \(m\) is the object's mass and \(v\) is its velocity. In our exercise, this form of energy is what the block has initially as it slides across the table. As the block collides with the spring, this energy is transformed, playing a crucial role in the law of conservation of energy.
Elastic Potential Energy
Elastic potential energy is the energy stored in elastic materials as they are compressed or stretched. Springs are a common example, and the energy stored is associated with the displacement from the spring's equilibrium position. The equation for the elastic potential energy (\(PE\text{)\text{)}} in a spring is \(PE = \frac{1}{2}kx^2\), where \(k\) is the spring constant—indicating the stiffness of the spring—and \(x\) is the amount of compression or stretch. In the problem, when the block compresses the spring, its kinetic energy is converted into elastic potential energy.
Spring Constant
The spring constant, denoted as \(k\), measures a spring's resistance to being compressed or stretched. It's a defining characteristic of the spring's physical properties and is expressed in units of force per unit length (\(N/m\)). A higher spring constant means a stiffer spring, requiring more force to achieve the same displacement compared to a spring with a lower constant. In our physics problem, the spring constant relates directly to the elastic potential energy stored in the spring when it is compressed by the block.
Law of Conservation of Energy
The law of conservation of energy states that energy cannot be created or destroyed in an isolated system; it can only be transformed from one form to another. This principle applies to the block-and-spring system from our exercise. Initially, all the system's energy is in the form of the block's kinetic energy. After the collision with the spring and the subsequent compression, that kinetic energy is transformed into elastic potential energy within the spring. To determine the compression of the spring, we set the initial kinetic energy equal to the final elastic potential energy and solve for the compression distance, as shown in the textbook solution.

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Most popular questions from this chapter

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