Final answer:
The equation U = 1/2(kh²) + mgh is solved for h by first subtracting mgh from both sides, then multiplying by 2/k, and finally taking the square root to isolate h, resulting in h = √[2(U - mgh)/k].
Step-by-step explanation:
The question involves solving for the variable h in the equation U = 1/2(kh²) + mgh. Here, U represents the total mechanical energy, k is the spring constant (in the potential energy stored in a spring), and mg represents the mass times the gravitational acceleration which is a part of the gravitational potential energy (G.P.E.). Solving for h requires isolating h on one side of the equation. First, subtract the term mgh from both sides to isolate the spring potential energy term:
1/2(kh²) = U - mgh
Then, multiply both sides by 2/k to solve for h²:
h² = 2(U - mgh)/k
Finally, take the square root of both sides to find h:
h = √[2(U - mgh)/k]
This will give us the value of h. Depending on the values of U, m, g, and k which are known from the context or given in the problem, these steps will provide the numerical solution for h.