Why Do Different Methods of Loading Affect Spring Calculations?

[confusseed]

can someone explain to me how springs work?
I already know the equation of elastic pe= 1/2kx^2 but I need more explanation!
how come when a mass is dropped from a hanging spring I can multiply the mass by 9.8m/s^2 to get the force, then divide the answer by the spring's constant (20n/cm) then divide by two and get the right answer every time but when it is gently let down, I get the right answer without dividing by two and how come these methods do not work when the spring is on the ground and the mass is being dropped from a height?

 

[qspeechc]

Your making me confused! I think you should try expressing yourslef more clearly, and people will be better equipted to help you!

Ok, for the P.E. I do not know how in-depth you go into this stuff. PE = -INT(F), that is, the potential energy is ALWAYS the negative integral of the force. Since the force for a spring is -kx, it is apparent where the PE equation comes from. We expect it to be positive all the time, because for all displacements (stretching of the spring) it must have a positive potential, as it stands to gain kinetic energy. Also note for x=o the potential is zeo, as we expect.

I think you want to find the displacement of a mass hanging on a spring. Well, first note that there are two forces acting on the mass. Gravitational force acting down, and th spring acting upwards. Therefore:
-kx = mg

mg is the force due to gravity, and -kx the force due to the spring. We equate them because when the block comes to rest, the forces must be equal.

Solving for x:
x = -mg/k

The negative sign just tells us it is displaced downwards.
I don't know what else you want.

 

[ogb p]

Sure, I would be happy to explain how springs work. A spring is a flexible object that can be stretched or compressed when a force is applied to it. The force applied to the spring causes it to store potential energy, known as elastic potential energy, which is given by the equation you mentioned, PE= 1/2kx^2. In this equation, k represents the spring constant, which is a measure of how stiff or flexible the spring is, and x represents the displacement or change in length of the spring.

When a mass is dropped from a hanging spring, gravity is the force that is acting on the mass. As you correctly stated, you can use the equation F=mg to calculate the force due to gravity, where m is the mass and g is the acceleration due to gravity (9.8m/s^2). However, when the mass is attached to the spring, the force of gravity is not the only force acting on the mass. The spring is also exerting a force on the mass, known as the restoring force, which is given by F=-kx. This force is directed in the opposite direction of the displacement of the spring and is what causes the spring to return to its original length after being stretched or compressed. So, in order to calculate the total force acting on the mass, you need to add the force due to gravity and the restoring force of the spring, which gives you the correct answer when you divide by the spring constant and two.

When the mass is gently let down, the force due to gravity is still acting on the mass, but the spring is not being stretched as much, so the restoring force is smaller. This is why you don't need to divide by the spring constant and two in this situation, as the restoring force is not as significant.

When the spring is on the ground and the mass is dropped from a height, the force of gravity is still acting on the mass, but there is no restoring force from the spring because it is not attached to the mass. Therefore, the equation F=mg will give you the correct answer without the need to divide by the spring constant or two.

I hope this explanation helps to clarify how springs work and why the different methods give different results in different situations. Please let me know if you have any other questions or need further clarification.