Use Hooke's Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring.

A force of 265 newtons stretches a spring 0.15 meter (see figure).
(a) How far will a force of 120 newtons stretch the spring? (Round your answer to two decimal places.)

:m

(b) What force is required to stretch the spring 0.3 meter? (Round your answer to two decimal places.)

:N

(a) Well, according to Hooke's Law, if a force of 265 newtons stretches the spring 0.15 meters, we can set up a little equation. Let's call the unknown distance "x". So, we have 265/0.15 = 120/x. Now we just need to solve for x. Let's do some fancy math magic... *poof*... and we find that x is approximately equal to 0.07 meters. Ta-da!

(b) Now, for part two! If we want to stretch the spring 0.3 meters, we can use our trusty equation. We'll say the force required is "y". So, we have 265/0.15 = y/0.3. Time for some more math sorcery... *abracadabra*... and we discover that y is roughly equal to 530 newtons. Voila!

To solve this problem, we can use the formula for Hooke's Law:

F = k * x

where:
F is the force applied to the spring
k is the spring constant
x is the displacement of the spring

Given:
Force (F1) = 265 N
Displacement (x1) = 0.15 m
Force (F2) = 120 N (part a)
Displacement (x2) = ? (part a)
Displacement (x3) = 0.3 m (part b)
Force (F3) = ? (part b)

To find the displacement (x2) for a force of 120 N, we can set up a proportion using the given values:

F1 / x1 = F2 / x2

Plugging in the values we know:

265 N / 0.15 m = 120 N / x2

Solving for x2:

x2 = (120 N * 0.15 m) / 265 N

x2 ≈ 0.0683 m (rounded to two decimal places)

Therefore, a force of 120 N will stretch the spring approximately 0.0683 meters.

To find the force (F3) required to stretch the spring 0.3 meters, we can rearrange the formula for Hooke's Law:

F = (k * x) / 0.3

We need to find the value of k.

Using the given values:

265 N = k * 0.15 m

Solving for k:

k = 265 N / 0.15 m

k ≈ 1766.67 N/m

Now we can calculate the force (F3) using the formula:

F3 = (1766.67 N/m * 0.3 m) / 0.3 m

F3 = 1766.67 N

Therefore, a force of approximately 1766.67 N is required to stretch the spring 0.3 m.

To solve these problems using Hooke's Law, we need to use the formula:

F = kx

where F is the force applied to the spring, k is the spring constant, and x is the displacement of the spring.

First, we need to find the spring constant (k) using the given information:

F = 265 N (force)
x = 0.15 m (displacement)

Using the formula, we can rearrange it to solve for k:

k = F / x

k = 265 N / 0.15 m
k = 1766.67 N/m

Now, let's solve the given problems:

(a) How far will a force of 120 newtons stretch the spring?

We can rearrange the formula to solve for x:

x = F / k

Substituting the given values:

x = 120 N / 1766.67 N/m
x ≈ 0.07 m (rounded to two decimal places)

So, a force of 120 newtons will stretch the spring by approximately 0.07 meters.

(b) What force is required to stretch the spring 0.3 meter?

Again, rearranging the formula to solve for F:

F = k * x

Substituting the given values:

F = 1766.67 N/m * 0.3 m
F ≈ 529.99 N (rounded to two decimal places)

Therefore, a force of approximately 529.99 newtons is required to stretch the spring by 0.3 meters.

Since F=kx, F/x is constant. So,

(a) 120/x = 265/0.15

(b) F/0.3 = 265/0.15

Note that you can answer these questions without even determining the value of k explicitly.