Problem 77 - College Physics

Question

\(\bullet\) Bumper guards. You are asked to design spring bumpers for the walls of a parking garage. A freely rolling 1200 \(\mathrm{kg}\) car moving at 0.65 \(\mathrm{m} / \mathrm{s}\) is to compress the spring no more than 7.0 \(\mathrm{cm}\) before stopping. (a) What should be the force constant of the spring, and what is the maximum amount of energy that gets stored in it? (b) If the springs that are actually delivered have the proper force constant but can become compressed by only \(5.0 \mathrm{cm},\) what is the maximum speed of the given car for which they will provide adequate protection?

Step-by-Step Solution

Answer

The spring constant should be 10379.59 N/m, with 253.5 J stored energy. Maximum car speed should be 0.147 m/s for 5 cm compression.

Step 1: Understanding the Problem

We need to design a spring that can stop a 1200 kg car moving at 0.65 m/s by compressing up to 7.0 cm. Part (a) asks for the spring's force constant and maximum energy stored, while part (b) asks for the maximum speed given a compression limit of 5.0 cm.

Step 2: Relevant Formulas

We will use the formula for kinetic energy, \( KE = \frac{1}{2}mv^2 \), and the elastic potential energy stored in a spring, \( PE = \frac{1}{2}kx^2 \), where \( k \) is the spring constant, \( m \) is mass, \( v \) is velocity, and \( x \) is compression.

Step 1: Calculate the Kinetic Energy

Using the car's mass and speed, calculate its initial kinetic energy:\[ KE = \frac{1}{2} \times 1200 \, \text{kg} \times (0.65 \, \text{m/s})^2 = 253.5 \, \text{J} \]

Step 2: Relate Kinetic Energy to Spring Potential Energy

The spring must store all the kinetic energy, so set \( PE = KE \). Thus, \( \frac{1}{2}kx^2 = 253.5 \). Use this to find \( k \) for \( x = 0.07 \, \text{m} \).

Step 3: Solve for Spring Constant 'k'

Plug the known values into the equation to find the spring constant:\[ k = \frac{2 \times 253.5}{(0.07)^2} = 10379.59 \, \text{N/m} \]

Step 4: Calculate Maximum Energy Stored

The maximum energy stored in the spring is equal to the initial kinetic energy, which we've found to be 253.5 J.

Step 5: Maximum Speed with a 5.0 cm Compression

Rearrange the potential energy formula for a spring to find the maximum speed for \( x = 0.05 \, \text{m} \), with the same \( k \): \[ KE = \frac{1}{2}k(0.05)^2 = 12.97 \, \text{J} \] Then solve for new speed with \( v = \sqrt{\frac{2KE}{m}} \): \[ v = \sqrt{\frac{2 \times 12.97}{1200}} = 0.147 \, \text{m/s} \]

Key Concepts

Elastic Potential Energy

Elastic potential energy is a form of energy stored within elastic materials as a result of their deformation. Consider a spring bumper in a parking garage used to stop a moving car. When the car hits the spring, the spring compresses and the kinetic energy from the car is transformed into elastic potential energy. The formula used to calculate this energy is:\[PE = \frac{1}{2}kx^2\]where:

• \(k\) is the spring constant, a measure of the spring's stiffness.
• \(x\) is the compression distance.

In our case, we need to design a spring that can store all of the car's kinetic energy by compressing up to 7.0 cm. If done successfully, the energy stored in the spring equals the car's initial kinetic energy, ensuring the car comes to a stop safely.

Kinetic Energy

Kinetic energy is the energy of motion. For the car moving in our parking scenario, the kinetic energy can be determined using its mass and velocity. The formula is:\[KE = \frac{1}{2}mv^2\]where:

• \(m\) is the car's mass.
• \(v\) is the car's velocity.

Before the car hits the spring, it has kinetic energy due to its motion. This energy needs to be fully transferred into the spring to ensure the car stops safely. In the example provided, the initial kinetic energy was calculated for a car moving at 0.65 m/s, resulting in 253.5 Joules. This energy is crucial for determining how much energy the spring must store to stop the car.

Spring Compression

Spring compression refers to how much a spring is compacted under a force. This measure is crucial in designing systems like the parking garage bumpers. In our scenario, the spring can compress by 7.0 cm, transferring the car's kinetic energy into stored elastic energy.

The degree of compression directly affects how much energy the spring can store. Using the spring compression formula, the spring constant needs to suit the maximum compression to prevent the car from moving any further. In our example, different scenarios were explored where one spring allowed only 5.0 cm of compression. This decreases the energy that can be stored, directly impacting the speed of the car that can be safely stopped.

Force Constant Calculation

The force constant, also known as the spring constant \(k\), indicates how resistant a spring is to compression or stretching. A higher force constant means a stiffer spring.

To calculate \(k\), we use the formula:\[k = \frac{2 imes KE}{x^2}\]It is derived by setting the elastic potential energy equal to the kinetic energy, ensuring energy conservation. In the task at hand, with a given kinetic energy and a maximum compression of 7.0 cm, the spring constant was calculated to be 10379.59 N/m.

This ensures that the spring can absorb all the car's kinetic energy, stopping it effectively in the garage. The calculation of \(k\) is critical because it guides the design of effective spring bumpers.