Problem 85 - University Physics with Modern Physics

Question

A 5.00 -kg block is moving at \(v_{0}=6.00 \mathrm{m} / \mathrm{s}\) along a frictionless, horizontal surface toward a spring with force constant \(k=500 \mathrm{N} / \mathrm{m}\) that is attached to a wall (Fig. P6.85). The spring has negligible mass. \begin{equation} \begin{array}{l}{\text { (a) Find the maximum distance the spring will be compressed. }} \\ {\text { (b) If the spring is to compress by no more than } 0.150 \mathrm{m}, \text { what }} \\\ {\text { should be the maximum value of } v_{0} ?}\end{array} \end{equation}

Step-by-Step Solution

Answer

(a) Maximum compression is 0.6 m. (b) Maximum initial velocity for 0.150 m compression is 2.74 m/s.

Step 1: Determine the Initial Energy of the System

The block's initial kinetic energy can be calculated using the formula for kinetic energy: \[ KE_i = \frac{1}{2}mv_0^2 \] where \( m = 5.00 \text{ kg} \) and \( v_0 = 6.00 \text{ m/s} \). Substitute these values into the equation to find \( KE_i \).

Step 2: Set Up the Energy Conservation Equation

Since there is no friction, energy is conserved. The initial kinetic energy of the block will be converted completely into the potential energy stored in the spring at maximum compression. The potential energy in the spring is given by: \[ PE_s = \frac{1}{2}kx^2 \] Equate the kinetic energy to the potential energy to find: \[ \frac{1}{2}mv_0^2 = \frac{1}{2}kx^2 \] Simplify to solve for \( x \), the compression of the spring.

Step 3: Solve for Maximum Compression

By solving the equation \( mv_0^2 = kx^2 \), we find:\[ x = \sqrt{\frac{mv_0^2}{k}} \] Substitute \( m = 5.00 \text{ kg} \), \( v_0 = 6.00 \text{ m/s} \), and \( k = 500 \text{ N/m} \) to solve for \( x \).

Step 4: Calculate Potential Energy for Given Compression

If the compression \( x \) is limited to \( 0.150 \text{ m} \), calculate the total stored energy using:\[ PE_s = \frac{1}{2}k(0.150)^2 \] This energy will equal the maximum possible kinetic energy of the block.

Step 5: Determine Maximum Velocity for Given Compression Limit

Using the maximum potential energy found in Step 4, solve for the maximum initial velocity \( v_0 \) using:\[ KE_i = \frac{1}{2}mv_{0_{max}}^2 = PE_s \] Solve for \( v_{0_{max}} \), keeping the compression limit in mind. Rearrange to find:\[ v_{0_{max}} = \sqrt{\frac{2PE_s}{m}} \] Substitute values to find \( v_{0_{max}} \).

Key Concepts

Kinetic Energy

Kinetic energy is all about motion, and it describes how much energy an object has due to its movement. Imagine a ball rolling down a hill or a car speeding along a highway. That's kinetic energy at play! The formula to calculate kinetic energy is:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass of the object, and \( v \) is its velocity.

In our exercise, we're looking at a 5.00-kg block moving with a velocity of 6.00 m/s. Substituting these values into the formula, we can find the initial kinetic energy of the block:

\[ KE_i = \frac{1}{2} \times 5.00 \times (6.00)^2 \] Calculating this gives us the initial kinetic energy, which serves as a starting point in our energy conservation analysis.

Potential Energy

Potential energy, on the other hand, is all about position. It measures the energy stored in an object because of its position or configuration. One of the common types of potential energy is the elastic potential energy found in springs.

For springs, the potential energy is given by:

• \( PE_s = \frac{1}{2}kx^2 \)

where \( k \) is the spring constant (a measure of the spring's stiffness), and \( x \) is the compression or extension of the spring.

In this exercise, as the 5.00-kg block moves towards the spring and compresses it, its kinetic energy is transformed into elastic potential energy. By setting the initial kinetic energy equal to the maximum potential energy, we can determine the maximum distance the spring will compress.

Spring Compression

Spring compression is the action of a spring being pressed into a more compact form, storing potential energy in its coils. When a moving object, like our 5.00-kg block, hits the spring, the spring shortens and stores energy, which can be calculated using the potential energy formula for springs.

To find the maximum compression of the spring, we apply energy conservation principles. Initially, all energy is kinetic. As the block compresses the spring, this kinetic energy turns into potential energy. This transformation is captured by the equation:

• \( \frac{1}{2}mv_0^2 = \frac{1}{2}kx^2 \)

Solving for \( x \), the maximum compression, involves substituting known values of mass, velocity, and spring constant into this equation:

\[ x = \sqrt{\frac{mv_0^2}{k}} \]Finally, with this formula, we can calculate how much the spring compresses when the block comes to a stop, effectively converting all its kinetic energy into stored potential energy.