Question

A railroad car of mass \(m\) is attached to a stop in a railroad yard. The stop consists of four identical metal rods of length \(L\), radius \(R\) and elastic modulus \(E\) that are arranged symmetrically and are fixed to a rigid wall at one end and welded to a rigid plate at the other. The plate is hooked to the stationary railroad car as shown. In a docking

Short answer

Answer: To find the final velocity (vf) of the combined cars, follow these steps: 1. Calculate the initial kinetic energy (K.E.) of the second car: \(K.E. = \frac{1}{2}mv_{1}^{2}\) 2. Determine the spring constant k of a single rod: \(k = \frac{EA}{L}\) and \(A = \pi R^2\) 3. Calculate the equivalent spring constant of the four-rod system: \(k_{eq} = 4k\) 4. Find the maximum compression x in the rods by equating the potential energy in the rods to the initial kinetic energy of the second car: \(P.E. = \frac{1}{2}k_{eq}x^2 = K.E.\) 5. Determine the final velocity vf using conservation of momentum: \(mv_1 = 2mv_f\), solve for vf. By following these steps and substituting the given values, you can calculate the final velocity (vf) of the two-car system after the energy absorption by the rods.

Step-by-step solution

Calculate the initial kinetic energy of the second railroad car

The initial kinetic energy (K.E.) of the second railroad car can be determined using the equation: \(K.E. = \frac{1}{2}mv_{1}^{2}\). Substitute the mass \(m\) and initial velocity \(v_1\) of the second railroad car to calculate the initial kinetic energy.

Determine the spring constant k of a single rod.

To determine the spring constant of a single metal rod, we can use the formula: \(k = \frac{EA}{L}\), where \(E\) is the elastic modulus, \(L\) is the length of the rod, and \(A\) is the cross-sectional area of the rod (with radius \(R\)). The cross-sectional area \(A\) can be calculated using the formula: \(A = \pi R^2\). After calculating \(A\), substitute the values of \(E\), \(A\), and \(L\) to find the spring constant of a single rod.

Determine the equivalent spring constant of the system of rods.

The four rods are arranged symmetrically. Since they are working together in parallel, their spring constants add up, giving us the equivalent spring constant of the system: \(k_{eq} = 4k\). Substitute the spring constant of a single rod obtained in Step 2 to find the equivalent spring constant of the system.

Calculate the maximum compression x in the rods.

Since the entire kinetic energy of the second railroad car is absorbed in the form of potential energy in the rods, we can equate the maximum potential energy in the rods to the initial kinetic energy of the second car. The potential energy in a spring system can be given by the formula: \(P.E. = \frac{1}{2}k_{eq}x^2\). Equating P.E. to the initial kinetic energy calculated in Step 1, we can find the maximum compression x in the rods.

Determine the response of the two-car system after docking.

After the rods have absorbed the entire kinetic energy of the second car, the two-car system comes to a standstill for a brief moment. However, the compressed rods now exert a force on the combined cars, pushing them back and accelerating them to some velocity \(v_f\). To find \(v_f\), we can use conservation of momentum. Initially, the total momentum of the system was \(mv_1\). After docking, the total momentum of the system will be \(2mv_f\) (as there are now two cars). Equating the initial and final momenta, we can solve for the final velocity \(v_f\) of the combined railroad cars' system.

Key concepts

Elastic Modulus

Elastic modulus is a fundamental property of materials that describes how much they will stretch or compress when subjected to stress. It is often referred to as the "stiffness" of a material. In other words, it measures the resistance of a material to elastic deformation.
Elastic modulus is vital in mechanics and materials science, as it helps in predicting how materials will react under different forces.
For a metal rod in our exercise, the elastic modulus is represented by the symbol

• E. The formula used to relate the elastic modulus (E) to the stress and strain is \( \sigma = E\varepsilon \). Here, \(\sigma\) is stress, and \(\varepsilon\) is strain.

This equation tells us that if we know the elastic modulus and the amount of strain, we can find the stress, or vice versa. Understanding elastic modulus is crucial in ensuring materials will serve their purpose without failure in structural applications.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. In this exercise, it's essential to calculate the initial kinetic energy of the railroad car, as this energy is converted into potential energy in the rods during compression.
Kinetic energy is given by the formula:

• \(K.E. = \frac{1}{2}mv^{2}\). Here, \(m\) is the mass, and \(v\) is the velocity of the object.

When dealing with kinetic energy, it's important to understand that it depends on both the mass and the square of the velocity. This means that an object with a larger mass or higher velocity will have significantly more kinetic energy.
Knowing how to calculate kinetic energy allows us to understand how energy is transferred between objects in motion, such as the railway cars in our exercise. This energy transfer is crucial in analyzing the compression of the rods when the car stops abruptly.

Spring Constant

The spring constant is a coefficient that indicates the stiffness of a spring or rod in terms of how much force it takes to stretch or compress it. In our exercise involving railroad cars and metal rods, the spring constant plays a significant role in analyzing how the rods respond to the forces exerted on them by the car.
The spring constant of a single rod is determined using the formula:

• \(k = \frac{EA}{L}\), where \(E\) is the elastic modulus, \(A = \pi R^2\) is the cross-sectional area, and \(L\) is the length of the rod.

The cross-sectional area \(A\) is crucial because it affects how spread out the force is over the material. Additionally, since multiple rods are used, the equivalent spring constant of the entire set-up is calculated by summing up the spring constants of individual rods:

• \(k_{eq} = 4k\).

Understanding the spring constant is essential for predicting how materials will behave under stress and how much they will compress or extend when force is applied.

Conservation of Momentum

Conservation of momentum is a principle that states the total momentum of an isolated system remains constant if no external forces act upon it. This fundamental concept of physics allows us to solve problems involving collisions and other interactions between objects.
In our exercise, after the kinetic energy is absorbed, it's crucial to use conservation of momentum to determine the movement of the two-car system after they dock.

• Initially, the system's total momentum is \(mv_1\), where \(m\) is the mass of the moving car and \(v_1\) is its velocity.
• After docking, the momentum becomes \(2mv_f\), since the cars move together with a final velocity \(v_f\).
• By setting the initial and final momentum equal (\(mv_1 = 2mv_f\)), we can solve for the final velocity \(v_f\).

Understanding momentum conservation helps analyze how objects will behave in interactions, ensuring calculations in dynamics are accurate.