Final answer:
To find the spring constant when the mass's total energy is evenly split between kinetic energy and spring potential energy, we can use the equation U = 1/2kx^2. By rearranging the equation and plugging in the given values, we can determine the spring constant.
Step-by-step explanation:
To find the spring constant, we need to use the equation for potential energy stored in a spring, which is given by U = 1/2kx^2. Here, U is the potential energy, k is the spring constant, and x is the displacement from the resting position. In this case, the total energy is evenly split between kinetic energy and spring potential energy, which means the kinetic energy is equal to the potential energy. Therefore, we can set the equation for kinetic energy, 1/2mv^2, equal to 1/2kx^2 and solve for k.
Let's use the given information that the mass is 15 cm from its resting position. Since x is the displacement, we need to convert cm to m, which gives us x = 0.15 m. Substituting this into the equation, we get 1/2mv^2 = 1/2k(0.15)^2. We can then rearrange the equation to solve for k.
k = (1/2mv^2)/(0.15)^2 = (1/2 * m * (v^2))/(0.15^2)
Now we can plug in the values for mass, velocity, and the conversions to find the spring constant.