Why is a spring a conservative force?

• OmegaKV
In summary, the amount of work done on the bowling ball will be different depending on whether it is dropped directly onto point P or if it is dropped onto a spring first and then onto point P. This is because the paths are different and different forces are involved, such as the force of gravity and the force of the spring. The work done on the ball by the floor must also be taken into account, as it acts as a spring with its own spring constant. Ultimately, the total amount of work will still equal mgh for conservation of energy.
OmegaKV
Let's say you are holding a bowling ball in the air, and there is a spring on the ground directly below it. You drop the bowling ball and it lands on the spring, compressing it, and then the bowling ball rolls off the spring onto a point P right next to the spring. Wouldn't this do a different amount of work than if you were to just throw the bowling ball directly onto P, bypassing the spring?

You will have to be more specific. What would do a different work on what?

Simon Bridge
OmegaKV said:
Wouldn't this do a different amount of work than if you were to just throw the bowling ball directly onto P, bypassing the spring?
i agree with the above comment -that more details are reqired
but as it stands the amount of work will be different as the paths are different and we do not know what type of forces are operating in the next throw- as far as spring action is concerned if it is ideal spring the motion will be holding the conservation of energy.

Orodruin said:
You will have to be more specific. What would do a different work on what?

The work done on the bowling ball by gravity and the spring.

Here are two ways to get to P with ball initially at height h:

Method 1 (bypass the spring):
Step 1) Move the ball horizontally so that it is positioned above P (0 work)
Step 2) Drop the ball (Work = mgh)
Total work on ball = mgh

Method 2 (drop ball on spring):
Step 1) Drop the ball, and the ball lands on the spring. If the uncompressed spring is y tall, the work from this step is: mg(h-y) plus the integral of (mg-ky) from 0 to y.
Step 2) Move the ball horizontally onto P (0 work if floor is frictionless).
Total work on the ball = mg(h-y)+mgy-(1/2)ky^2=mgh-(1/2)ky^2

You are missing the work done by the floor on the ball in order to stop it in the case where it is not dropped on the spring. There is nothing strange going on here.

Orodruin said:
You are missing the work done by the floor on the ball in order to stop it in the case where it is not dropped on the spring. There is nothing strange going on here.

What if the spring is tight enough that the ball never completely hits the ground, and then you push the ball off the spring right when the ball's velocity reaches 0 (i.e. right before the ball starts going back up)?

OmegaKV said:
What if the spring is tight enough that the ball never completely hits the ground, and then you push the ball off the spring right when the ball's velocity reaches 0 (i.e. right before the ball starts going back up)?
This does not change the fact that the force from the floor will do work in the other case.

OmegaKV said:
Wouldn't this do a different amount of work than if you were to just throw the bowling ball directly onto P, bypassing the spring?

Either way you are dealing with a generalized spring. Take the case where the ball is dropped directly on to the floor. If the floor is undamaged and left without plastic deformation, then it too acted as a spring that stopped the ball. Real materials, like a floor, have some level of elasticity. A floor simply has a different spring constant (elastic modulus) than the actual spring.

In Method 1 the rebound of the spring must be accounted for. In Method 2 the floor never does any work unless it deflects (work = force X distance). For that calculation you would need the floor stiffness characteristics (another spring!). In the end, the available energy is in mgh. Without getting into the weeds and tiny details the end result has to add up to mgh for conservation of energy.

1. What is a conservative force?

A conservative force is a type of force that does not depend on the path taken by an object but only on its initial and final positions. This means that the work done by a conservative force is independent of the path taken by the object and only depends on its starting and ending positions.

2. How does a spring exhibit conservative force?

A spring exhibits conservative force because the work done by the spring is independent of the path taken by the object attached to it. The force exerted by a spring is directly proportional to its displacement, which means that the work done is the same whether the object is stretched or compressed.

3. Why is a spring considered a conservative force?

A spring is considered a conservative force because it follows the law of conservation of energy. When an object is attached to a spring, it gains potential energy due to its displacement. This potential energy can be converted back into kinetic energy, and the total energy remains constant.

4. How does the conservation of energy apply to a spring?

The conservation of energy applies to a spring because the work done by the spring is stored as potential energy in the object attached to it. When the object is released, the potential energy is converted back into kinetic energy, and the total energy remains constant. This means that the work done by the spring is equal to the change in potential energy of the object.

5. What are some examples of other conservative forces?

Some examples of other conservative forces include gravity, electrostatic force, and magnetic force. These forces also do not depend on the path taken by an object and only depend on its initial and final positions. They follow the law of conservation of energy and can be represented by a potential energy function.

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