# Difference between hooke's law and the work done in a spring?

• dnt
In summary, Hooke's Law states that the spring force is equal to -kx, but the work done in stretching a spring is equal to 1/2 kx^2. This is because work is not equal to Fd when the force is not constant, as is the case with a spring. The correct equation for work in this case is W = ∫Fdx, which, when plugged in with F = -kx, results in W = -kx^2. The 1/2 in the equation 1/2kx^2 comes from the integration process when calculating work. It is important to note that this equation is only valid for non-constant forces, such as a spring.
dnt
hooke's law says the spring force = -kx

but the work done in stretching a spring = 1/2 kx^2

isnt work = Fd?

so using W=Fd and F=-kx (hooke's law) shouldn't the work come out to:

W = -kx^2? (arent d and x the same? both are distance stretched)

where does the 1/2 come from in the equation 1/2kx^2?

can someone help clarify the difference and when to use each equation. thanks.

dnt said:
hooke's law says the spring force = -kx

but the work done in stretching a spring = 1/2 kx^2

isnt work = Fd?

so using W=Fd and F=-kx (hooke's law) shouldn't the work come out to:

W = -kx^2? (arent d and x the same? both are distance stretched)

where does the 1/2 come from in the equation 1/2kx^2?

can someone help clarify the difference and when to use each equation. thanks.
W = Fd is only correct when the force is constant and in the direction of a displacement of distance d. The force is not constant for a spring. Have you taken calculus?

dnt said:
isnt work = Fd?
That is correct, provided the force is constant. However, generally work can be expressed thus;

$$W=\int^{x_{1}}_{x_{0}}\; F \cdot dx$$

Edit: Too slow again

Hootenanny said:
That is correct, provided the force is constant. However, generally work can be expressed thus;

$$W=\int^{x_{1}}_{x_{0}}\; F \cdot dx$$

Edit: Too slow again

...which, after plugging in F = - k x, answers your question.

## 1. What is Hooke's Law?

Hooke's Law is a principle in physics that states that the force needed to extend or compress a spring is directly proportional to the distance the spring is stretched or compressed from its rest position. It is named after the scientist Robert Hooke who first described this relationship in the 17th century.

## 2. How is Hooke's Law related to the work done in a spring?

Hooke's Law is directly related to the work done in a spring. According to Hooke's Law, the force applied to a spring is proportional to the distance it is stretched or compressed. This means that the work done in stretching or compressing a spring is also proportional to the distance. This relationship is represented by the equation W = 1/2kx^2, where W is the work done, k is the spring constant, and x is the distance the spring is stretched or compressed.

## 3. What is the difference between Hooke's Law and the work done in a spring?

The main difference between Hooke's Law and the work done in a spring is that Hooke's Law describes the relationship between the force applied to a spring and the distance it is stretched or compressed, while the work done in a spring is a measure of the energy required to stretch or compress the spring. Hooke's Law is a principle that can be used to calculate the force needed to stretch or compress a spring, while the work done in a spring is a measure of the energy transfer that occurs as a result of this force.

## 4. How is Hooke's Law applied in real-life situations?

Hooke's Law is applied in many real-life situations, such as in the design of springs for various mechanical devices. For example, car suspensions and pogo sticks both use Hooke's Law to function properly. It is also used in the construction of buildings and bridges to determine the amount of force that can be applied to structures without causing damage. Additionally, Hooke's Law is used in medical devices such as prosthetics and orthotics to design and test the materials used to create these devices.

## 5. Are there any limitations to Hooke's Law?

While Hooke's Law is a useful principle, it does have some limitations. It assumes that the spring is being stretched or compressed within its elastic limit, meaning that the spring will return to its original shape once the force is removed. If the force applied to the spring is too great, it may exceed its elastic limit and permanently deform or break. Additionally, Hooke's Law only applies to linear springs, meaning that the force and displacement must be in the same direction. Non-linear springs, such as those used in clocks and watches, do not follow Hooke's Law.

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