An unstretched spring is 11.0 cm long. When you hang an 590 g weight from it, it stretches to a length of 14.0 cm. what is the force constant in (N/m) of this spring? what is the total mass must you hang from the spring to stretch it to a total length of 18cm?

x = 14-11 = 3 cm = .03 m

F = m g = .59kg * 9.81 = 5.79 N
k = F/x = 5.79/.03 = 193 n/m

F = 193 (.18-.11) = 13.5 N
m = F/g = 13.5/9.81 = 1.38 kg = 1380 g

Oh spring, why do you have to be so stretchy? Let's solve this math problem before you bounce away!

To find the force constant (k) of the spring, we can use Hooke's law. Hooke's law states that the force applied to a spring is directly proportional to the displacement (or stretch) of the spring.

First, let's find the change in length of the spring when the weight is added:

Change in length = Final length - Initial length
Change in length = 14.0 cm - 11.0 cm
Change in length = 3.0 cm

Now, let's find the force constant (k) using Hooke's law:

F = -k * x

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

In this case, the force applied is the weight of the object (590 g) acting downwards. We can convert the weight to Newtons using the formula:

Weight (in N) = mass (in kg) * acceleration due to gravity (9.8 m/s^2)

Weight = 0.59 kg * 9.8 m/s^2
Weight = 5.782 N

Now, let's plug in the values into Hooke's law:

5.782 N = -k * 0.03 m

Simplifying the equation:

k = -5.782 N / 0.03 m
k ≈ 192.733 N/m

So the force constant of the spring is approximately 192.733 N/m. Now, onto the next question!

To find the total mass you must hang from the spring to stretch it to a total length of 18 cm, we need to use the force constant we just found.

First, let's find the change in length again:

Change in length = Total length - Initial length
Change in length = 18.0 cm - 11.0 cm
Change in length = 7.0 cm

Now, let's find the total mass needed using Hooke's law:

F = -k * x

In this case, we want to find the force applied, so we rearrange the equation:

F = k * x

Let's plug in the values:

F = 192.733 N/m * 0.07 m
F ≈ 13.491 N

Now, to convert the force to mass, we use the formula:

Weight (in N) = mass (in kg) * acceleration due to gravity (9.8 m/s^2)

Weight = mass * 9.8 m/s^2
mass = Weight / 9.8 m/s^2

Using the weight we found:

mass = 13.491 N / 9.8 m/s^2
mass ≈ 1.376 kg

So, you'll have to hang a total mass of approximately 1.376 kg to stretch the spring to a length of 18 cm. Now, go grab some weights and have fun bouncing around!

To find the force constant of the spring, you can use Hooke's Law:

F = k * x

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

In this case, when the weight of 590 g (0.59 kg) is hung from the spring, the displacement is 14.0 cm - 11.0 cm = 3.0 cm = 0.03 m.

Now, we can substitute the values into Hooke's Law and solve for the force constant:

F = k * x
0.59 kg * 9.81 m/s^2 = k * 0.03 m

k = (0.59 kg * 9.81 m/s^2) / 0.03 m
k ≈ 192.32 N/m

Therefore, the force constant of this spring is approximately 192.32 N/m.

To find the total mass required to stretch the spring to a length of 18 cm, we can use the same formula and determine the force needed. Once we have the force, we can divide it by the acceleration due to gravity (9.81 m/s^2) to find the mass.

The displacement is 18 cm - 11 cm = 7 cm = 0.07 m.

F = k * x
F = 192.32 N/m * 0.07 m
F ≈ 13.46 N

Now, we can solve for the mass:

F = m * g
13.46 N = m * 9.81 m/s^2

m = 13.46 N / 9.81 m/s^2
m ≈ 1.37 kg

Therefore, you would need to hang approximately 1.37 kg to stretch the spring to a total length of 18 cm.

To find the force constant of the spring, 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.

Hooke's Law can be written as:
F = -kx

Where F is the force exerted by the spring, k is the force constant, and x is the displacement from the equilibrium position.

We can start by finding the force constant (k) using the given information. The unstretched length of the spring (x₁) is 11.0 cm, and the stretched length (x₂) is 14.0 cm.

First, let's convert the lengths to meters:
x₁ = 11.0 cm = 0.11 m
x₂ = 14.0 cm = 0.14 m

The displacement (x) is given by:
x = x₂ - x₁ = 0.14 m - 0.11 m = 0.03 m

The force (F) is determined by the weight suspended from the spring:
F = mg

Where m is the mass and g is the acceleration due to gravity (9.8 m/s²).

Given that the weight (F) is 590 g, we can convert it to kilograms:
m = 590 g = 0.59 kg

Substituting these values into Hooke's Law equation:
mg = kx
0.59 kg * 9.8 m/s² = k * 0.03 m

Solving for k:
k = (0.59 kg * 9.8 m/s²) / 0.03 m

Calculating k:
k = 192.67 N/m (rounded to two decimal places)

Therefore, the force constant (k) of the spring is approximately 192.67 N/m.

Now, let's calculate the total mass needed to stretch the spring to a length of 18 cm (0.18 m).

The displacement (x) is given by:
x = x₃ - x₁ = 0.18 m - 0.11 m = 0.07 m

Using the force constant (k) we calculated earlier:
F = kx

Substituting known values:
F = 192.67 N/m * 0.07 m

Calculating F:
F = 13.49 N

To find the mass (m), we can rearrange the equation F = mg:
m = F / g

Substituting known values:
m = 13.49 N / 9.8 m/s²

Calculating m:
m = 1.38 kg (rounded to two decimal places)

Therefore, the total mass needed to stretch the spring to a length of 18 cm is approximately 1.38 kg.