Problem 79 - University Physics with Modern Physics

Question

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 0.090 \(\mathrm{m}\) before stopping. What should be the force constant of the spring? Assume that the spring has negligible mass.

Step-by-Step Solution

Answer

The force constant of the spring should be approximately 62593 N/m.

Step 1: Determine the car's kinetic energy

The kinetic energy (KE) of the car can be calculated using the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the car (1200 kg) and \( v \) is its velocity (0.65 m/s).

Step 2: Calculate the car's kinetic energy

Substitute \( m = 1200 \) kg and \( v = 0.65 \) m/s into the formula: \( KE = \frac{1}{2} \cdot 1200 \cdot (0.65)^2 \). This gives \( KE \approx 253.5 \) J (Joules).

Step 3: Relate kinetic energy to spring compression

The kinetic energy of the car is converted completely into the potential energy stored in the spring when the car stops. The potential energy (PE) stored in a spring is given by \( PE = \frac{1}{2}kx^2 \), where \( k \) is the spring constant and \( x \) is the maximum compression of the spring (0.090 m).

Step 4: Find the spring constant

Since the kinetic energy and the spring's potential energy are equal when the car stops, set \( KE = PE \). Thus, \( 253.5 = \frac{1}{2}k(0.090)^2 \). Solve for \( k \): \( k = \frac{2 \times 253.5}{(0.090)^2} \).

Step 5: Calculate the spring constant

Evaluating the expression, \( k = \frac{2 \times 253.5}{0.0081} \approx 62592.59 \) N/m. Hence, the force constant of the spring should be approximately \( 62593 \) N/m.

Key Concepts

Understanding Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It's an important concept when dealing with moving vehicles like cars. The greater the speed or mass of the car, the higher its kinetic energy.
This energy can be calculated using the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) represents the mass of the car, and \( v \) is its velocity.
In our physics problem, we have a car with a mass of 1200 kg moving at 0.65 m/s. By substituting these values, the kinetic energy is found to be approximately 253.5 Joules.

• The "\( \frac{1}{2} \)" in the formula is a constant, ensuring that the energy calculation is dimensionally accurate.
• Kinetic energy is always positive, as both mass and the square of velocity are positive.
• Understanding kinetic energy helps in predicting the behavior of moving objects when they interact with other forces, like springs in this scenario.

Exploring Potential Energy in Springs

Potential energy in a spring is a form of stored energy, dependent on an object's position, specifically, how much the spring is compressed or stretched.
For springs, potential energy is determined by the formula \( PE = \frac{1}{2}kx^2 \), where \( k \) is the spring constant and \( x \) is the extent of the spring's compression or extension.
In the car and spring example, as the car comes to a stop, its kinetic energy is converted into potential energy stored in the compressed spring. This energy conversion is crucial because it determines the car's ability to stop using spring bumpers.

• "Spring constant" \( k \) is a measure of the spring's stiffness. Higher values mean a stiffer spring.
• If kinetic energy converts fully to spring potential energy, the car stops safely. Otherwise, extra force or distance might be required.
• The symmetry of potential and kinetic energy equations allows precise calculations in practical scenarios like parking garage bumpers.

Physics Problem Solving Approach

Solving physics problems methodically helps in understanding complex concepts easily. It often involves breaking down a problem into simpler parts.
Let's take a look at the chain of ideas: You start with known values, like mass and speed, to find kinetic energy. Then, relate it to the unknown, like the spring's constant, by using energy conversion principles.
For this car problem, follow these simplified steps:

• Identify what's given and determine necessary formulas, such as kinetic and potential energy.
• Calculate initial kinetic energy as a basis for further changes.
• Relate kinetic to potential energy, since conversion occurs when the car stops.
• Set up equations and solve for the unknown—here, the spring constant \( k \).
• Verify your solution with realistic units and estimated effects. This ensures the solution makes sense practically.

This organized approach prevents mistakes and enhances understanding.

The Role of Spring Compression

Spring compression in this context refers to how much the spring is compressed by the car before the car comes to a stop.
It is measured by how the distance (\( x \)) relates directly to the force exerted on or by the spring and indirectly to the spring's potential energy.
In our exercise, the spring is allowed to compress up to 0.090 meters only.

• Spring compression is proportional to the car's kinetic energy being absorbed as potential energy.
• The formula \( PE = \frac{1}{2}kx^2 \) guides the design of spring bumpers so that compression does not exceed physical device limits.
• Understanding spring compression helps engineers design safety devices that mitigate impact, reducing structural and occupant harm.

By carefully calculating compression, you can predict how the car stops safely without overshooting the mechanical limits of the springs.