Problem 73 - University Physics with Modern Physics

Question

You are asked to design spring bumpers for the walls of a parking garage. A freely rolling 1200-kg car moving at 0.65 m/s is to compress the spring no more than 0.090 m before stopping. What should be the force constant of the spring? Assume that the spring has negligible mass.

Step-by-Step Solution

Answer

The spring constant should be approximately 62,333 N/m.

Step 1: Identify the Problem

We need to find the force constant of the spring, also known as the spring constant, required to stop a car moving at a specific speed within a certain compression distance.

Step 2: Understand the Energy Transformation

The car's kinetic energy will be transformed into potential energy stored in the spring. Use the principle of conservation of energy: Initial kinetic energy of the car = Potential energy stored in the spring.

Step 3: Write the Expression for Kinetic Energy

The kinetic energy (KE) of the car can be calculated using the formula: \( KE = \frac{1}{2}mv^2 \).Plugging in the given values: \( KE = \frac{1}{2} \times 1200\,\text{kg} \times (0.65\,\text{m/s})^2 \).

Step 4: Calculate the Car’s Kinetic Energy

Calculate the car's initial kinetic energy:\[ KE = \frac{1}{2} \times 1200 \times 0.65^2 = 253.5\,\text{J} \].

Step 5: Write the Expression for Spring Potential Energy

The potential energy stored in the spring at maximum compression is given by: \( PE = \frac{1}{2}kx^2 \), where \( k \) is the spring constant and \( x \) is the compression distance.

Step 6: Set Up the Equation for Energy Conservation

Since energy is conserved:\( \frac{1}{2}mv^2 = \frac{1}{2}kx^2 \).From Step 4, we have the kinetic energy as 253.5 J. We need to solve for \( k \).

Step 7: Solve for Spring Constant \( k \)

Substitute \( KE = 253.5\,\text{J} \) and \( x = 0.090\,\text{m} \) into the energy conservation equation:\( 253.5 = \frac{1}{2}k(0.090)^2 \).Solve for \( k \):\[ k = \frac{2 imes 253.5}{(0.090)^2} = 62,333.3\,\text{N/m} \] (approximately).

Step 8: Round and Conclude

Round the spring constant to a practical value:The spring constant required is approximately \( 62,333\,\text{N/m} \).

Key Concepts

Conservation of Energy

The conservation of energy principle is crucial in understanding how the energy transformations occur between different states of a system. This principle states that energy cannot be created or destroyed, but can only be transformed from one form to another. In the context of the parking garage problem, the moving car initially has kinetic energy due to its motion. As it hits the spring bumper, this kinetic energy is gradually converted into spring potential energy until the car stops completely.
The total energy remains constant, meaning that whatever energy the car had at the start as kinetic energy, it should equal the energy stored in the spring as potential energy after the car compresses the bumper. This transformation allows us to set up an equation and solve for the unknowns, such as the spring constant, using known quantities like the mass of the car, its velocity, and the maximum compression of the spring.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It depends on two main factors: the mass of the object and its velocity. The formula used to calculate kinetic energy is:

• Kinetic Energy ( KE ) = \( \frac{1}{2}mv^2 \)

where:

• \( m \) is the mass of the object in kilograms (kg), and
• \( v \) is the velocity of the object in meters per second (m/s).

In the exercise, a 1200-kg car is considered, moving at a speed of 0.65 m/s. Plugging these values into the kinetic energy formula gives the car's initial energy. This energy, calculated to be 253.5 joules, will be fully transferred into the spring's potential energy when the car stops. Understanding this concept helps us realize how energy depends not just on how fast an object is moving but also on its mass.

Spring Potential Energy

When a spring is compressed or stretched, it stores energy, known as potential energy. The amount of energy stored can be calculated using the spring potential energy formula:

• Potential Energy ( PE ) = \( \frac{1}{2}kx^2 \)

where:

• \( k \) is the spring constant, representing the stiffness of the spring, and
• \( x \) is the displacement from the spring's resting position, either stretched or compressed, measured in meters.

In our parking garage scenario, the displacement \( x \) is the maximum compression of the spring (0.090 m). The spring's potential energy at maximum compression will equal the kinetic energy that was initially possessed by the car, as per the conservation of energy principle. This relationship allows us to solve for \( k \), the spring constant, which is a measure of how much force it takes to compress or stretch the spring by a certain length.

Force Constant

The force constant, often denoted as \( k \), is a key parameter of a spring that indicates its stiffness. A larger value of \( k \) means a stiffer spring that requires more force to achieve the same degree of compression or stretch, compared to a spring with a lower \( k \).
In problems involving energy conservation, the force constant is crucial because it dictates how much potential energy a spring can store when it is compressed or stretched. By rearranging the energy conservation equation, the spring constant can be determined from known values of initial kinetic energy and the maximum compression distance \( x \).
This concept helps design practical applications, such as springs in car bumpers or even engineering designs, by providing essential information about how the system will behave under given physical constraints, for instance, in stopping a moving vehicle safely with minimal distance.