A spring has an unstretched length of 10 cm . It exerts a restoring force F when stretched to a length of 12 cm .

For what total stretched length of the spring is its restoring force 3F?

At what compressed length is the restoring force 2F?

I keep doing this but I do not think Im using the right math to solve this

F = kx

Doubling F means doubling x as k will not change.

1)16cm

2)6cm

To solve these problems, we can use Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. In equation form, Hooke's law is expressed as:

F = -kx

where F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position.

1. For what total stretched length of the spring is its restoring force 3F?
To find the total stretched length, we need to determine the displacement from the unstretched length. Let's denote the unstretched length as x₀, the stretched length as x₁, and the total stretched length as x. We can set up the following equation:

F = -k(x - x₀)

Since we know that F = 3F, we can substitute this into the equation:

3F = -k(x - x₀)

Dividing both sides by -k and rearranging the equation, we get:

x - x₀ = -3

Now, we can substitute the given values:

12 cm - 10 cm = -3

Simplifying, we have:

2 cm = -3

Since this is a negative value, it means that the spring needs to be compressed by 2 cm from its unstretched length of 10 cm. Therefore, the total stretched length of the spring is 10 cm - 2 cm = 8 cm.

2. At what compressed length is the restoring force 2F?
Similar to the previous question, in this case, we need to determine the displacement from the unstretched length. Denote the compressed length as x₁ and the compressed displacement as Δx. We can set up the following equation:

F = -k(Δx + x₀)

Again, substituting F = 2F, we have:

2F = -k(Δx + x₀)

Dividing both sides by -k and rearranging the equation, we get:

Δx + x₀ = -2

Now, let's substitute the given values:

Δx + 10 cm = -2

Simplifying, we have:

Δx = -12 cm

Since this is a negative value, it means that the spring needs to be compressed by 12 cm from its unstretched length of 10 cm. Therefore, the compressed length of the spring is 10 cm - 12 cm = -2 cm.

To solve these problems involving a spring's restoring force, we can use Hooke's Law. Hooke's Law states that the force exerted by a spring is directly proportional to the displacement or change in length of the spring from its equilibrium position.

Let's break down each problem step by step:

1. For what total stretched length of the spring is its restoring force 3F?

To solve this problem, we need to find the relationship between the stretched length of the spring and the restoring force. Hooke's Law can be written as:

F = k * x

where F is the restoring force, k is the spring constant, and x is the displacement or change in length.

Given that the spring has an unstretched length of 10 cm and exerts a restoring force F when stretched to a length of 12 cm, we can calculate the change in length (x):

x = 12 cm - 10 cm
x = 2 cm

We also know that the restoring force is F when the length increase is 2 cm. Now, we can set up a proportion to find the total stretched length when the restoring force is 3F:

(F/2 cm) = (3F/x)

To solve for x, we can cross-multiply and then solve for x:

3F * 2 cm = F * x
6F cm = F * x

Now, we cancel out the F on both sides of the equation:

6 cm = x

Therefore, the total stretched length of the spring when the restoring force is 3F is 6 cm.

2. At what compressed length is the restoring force 2F?

For the second problem, we need to find the compressed length of the spring when the restoring force is 2F.

Using the same equation, F = k * x, we can rearrange it to solve for x:

x = F / k

Let's assume that the change in length when the restoring force is 2F is y. So we can write the equation:

y = 2F / k

However, we do not know the value of k (the spring constant) in this case. Without this information, we cannot determine the exact compressed length of the spring.

To solve the problem fully, we need either the spring constant or additional information that helps us find the spring constant.