A block on a horizontal surface is placed in contact with a light spring with spring constant , as shown in figure 1. when the block is moved to the left so that the spring is compressed a distance from its equilibrium length, the potential energy stored in the spring-block system is . when a second block of mass is placed on the same surface and the spring is compressed a distance , as shown in figure 2, how much potential energy is stored in the spring compared to the original potential energy ? all frictional forces are considered to be negligible.

Question: A block on a horizontal surface is placed in contact with a light spring with spring constant , as shown in figure 1. when the block is moved to the left so that the spring is compressed a distance from its equilibrium length, the potential energy stored in the spring-block system is . when a second block of mass is placed on the same surface and the spring is compressed a distance , as shown in figure 2, how much potential energy is stored in the spring compared to the original potential energy ? all frictional forces are considered to be negligible.

In this blog post, we will explore the concept of potential energy in a spring-block system. We will use the following scenario as an example:

A block of mass m_1 is placed on a horizontal surface and attached to a light spring with spring constant k, as shown in Figure 1. The spring is initially at its equilibrium length. When the block is moved to the left by a distance x_1, the spring is compressed and the block-spring system gains some potential energy. We can calculate this potential energy using the formula:

U_1 = (1/2)kx_1^2

where U_1 is the potential energy stored in the system when the spring is compressed by x_1.

Now, suppose we add another block of mass m_2 on top of the first block, as shown in Figure 2. The spring is further compressed by a distance x_2, and the system gains more potential energy. How much more? We can use the same formula to find out:

U_2 = (1/2)k(x_1 + x_2)^2

where U_2 is the potential energy stored in the system when the spring is compressed by x_1 + x_2.

To compare the potential energy in both cases, we can subtract U_1 from U_2 and get:

U_2 - U_1 = (1/2)k(x_1 + x_2)^2 - (1/2)kx_1^2

= (1/2)k(x_1^2 + 2x_1x_2 + x_2^2 - x_1^2)

= (1/2)k(2x_1x_2 + x_2^2)

This expression tells us how much more potential energy is stored in the system when we add the second block and compress the spring further. It depends on both x_1 and x_2, as well as the spring constant k. If we know these values, we can plug them in and get a numerical answer.

For example, if m_1 = 0.5 kg, m_2 = 0.3 kg, k = 200 N/m, x_1 = 0.05 m, and x_2 = 0.02 m, then we have:

U_2 - U_1 = (1/2)(200)(0.05)(0.02) + (1/2)(200)(0.02)^2

= 0.25 J + 0.04 J

= 0.29 J

This means that adding the second block and compressing the spring by 0.02 m increases the potential energy stored in the system by 0.29 J.