I'm guessing you divided E by x expecting to get F. But the equation energy = force x displacement only works if the force is constant.tallwallyb said:On the energy part, I keep getting mg=(1/2)*x*k, which is contradictory to Hooke's law F=mg=kx. What is going on?
Hooke's Law itself is not contradictory, but confusion often arises because it describes the force exerted by the spring, not the energy stored in it. The apparent contradiction stems from misunderstanding the relationship between force and energy. Hooke's Law states that the force is proportional to the displacement (F = -kx), while the energy stored in the spring is proportional to the square of the displacement (U = 1/2 kx^2).
In Hooke's Law, the force exerted by the spring is directly proportional to the displacement (F = -kx). However, the energy stored in the spring is given by the equation U = 1/2 kx^2, which means it depends on the square of the displacement. This quadratic relationship explains why the energy grows more quickly than the force as the displacement increases.
The energy stored in a spring is the work done to compress or stretch it. Work is calculated as the integral of force over displacement. Since the force varies linearly with displacement (F = kx), integrating this force with respect to displacement (from 0 to x) results in the quadratic term, hence U = 1/2 kx^2. This accounts for the increasing amount of work needed to further displace the spring.
Certainly! The work done to compress or stretch a spring is the integral of the force over the displacement. Mathematically, this is expressed as:\[ W = \int_0^x F \, dx = \int_0^x kx \, dx \]Evaluating this integral, we get:\[ W = \int_0^x kx \, dx = \frac{1}{2}kx^2 \]This result shows that the energy stored in the spring (U) is given by U = 1/2 kx^2.
Someone might find Hooke's Law contradictory when considering spring energy because they might not recognize that force and energy are related but distinct concepts. Hooke's Law describes a linear relationship between force and displacement, while the energy stored in the spring depends on the square of the displacement. Misunderstanding this distinction can lead to confusion about why the equations for force and energy look different.