Question

Consider a system of two toy railway cars (i.e., frictionless masses) connected to each other by two springs, one of which is attached to the wall, as shown in the figure. Let x1 and x2 be the displacement of the first and second masses from their equilibrium positions. Suppose the masses are m1=4 kg and m2=2 kg, and the spring constants are k1=128 N/m and k2=64 N/m. Set up a system of second-order differential equations that models this situation. x"" =

Answer 1

Final answer:

To model the given situation, we need to set up a system of second-order differential equations. The equations can be derived by considering the forces acting on each mass. The first mass experiences forces from both springs, while the second mass only experiences a force from the second spring.



Step-by-step explanation:

To set up the system of second-order differential equations that models this situation, we need to consider the forces acting on each mass. The first mass (m1) experiences a force due to the first spring (k1) and a force due to the second spring (k2) through the displacement x1. The second mass (m2) experiences a force only through the displacement x2.

Using Hooke's law, the force exerted by the first spring on mass m1 is -k1x1, and the force exerted by the second spring on mass m1 is -k2(x1-x2). The force exerted by the second spring on mass m2 is k2(x1-x2).

Applying Newton's second law (F=ma) to each mass, we have:

m1x1'' = -k1x1 - k2(x1-x2)

m2x2'' = k2(x1-x2)