Calculating Spring Constant - Force or Energy?

[Cintdrix]

I've been being confused lately as to the 2 methods. The example I'm thinking of is when a weight of mass m is hung on a spring and it stretches x meters.

First of all, I know you can equate the spring force (kx) to the force of gravity (mg), to get
k = mg/x

But is it also possible to say that the gravitational potential energy lost (mgx) is equal to the energy gained by the spring (1/2 kx^2)? When I do this, I get a different k which is half the original k and probably wrong. How can you calculate K for this problem using energy?

 

[dsol]

The first method F = mg = kx is what you would use to find the spring constant k. The second equation is used for the dynamic problem "How fast is the mass moving at distance x?". To solve, equate the potential energy "lost" by the downward motion to the potential energy of the spring AND the kinetic energy of the object: E = mgx = (1/2 kx^2) + (1/2 mv^2)

 

[astrokreng]

it is important to understand the concept of energy conservation and how it applies to the calculation of the spring constant in this scenario. Both methods, using force and using energy, are valid ways to calculate the spring constant. However, they are based on different principles and therefore may yield different results.

When using the force method, we are looking at the equilibrium point of the spring, where the forces of gravity and the spring are balanced. This gives us the equation k = mg/x, where m is the mass, g is the gravitational acceleration, and x is the displacement of the spring. This method is based on Newton's Second Law, which states that the sum of all forces acting on an object is equal to its mass times its acceleration.

On the other hand, when using the energy method, we are looking at the potential energy stored in the spring when it is stretched. In this case, we can equate the gravitational potential energy (mgx) to the elastic potential energy (1/2 kx^2) of the spring. This method is based on the principle of energy conservation, which states that energy cannot be created or destroyed, but can only be converted from one form to another.

It is important to note that these two methods are not interchangeable and should not be used together in the same calculation. This can result in incorrect values for the spring constant. If you are using the energy method, you must consider only the potential energy stored in the spring, not the force of gravity acting on the mass.

In conclusion, both methods, using force and using energy, are valid ways to calculate the spring constant. However, they are based on different principles and should be used independently. It is important to understand the concept of energy conservation and how it applies to the calculation of the spring constant in order to obtain accurate results.