A spring has a length of 0.190 m when a 0.300 kg mass hangs from it, and a length of 0.750 m when a 2.80 kg mass hangs from it.

(a) What is the force constant of the spring?

(b) What is the unloaded length of the spring?

F=kx

.3*g=k(.190-u)
2.80g=k(.750-u) u is the unstretched length.

solve for k.

What is the unstretched length?

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 is given by the formula: 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.

(a) To find the force constant of the spring, we need to use the information provided for both masses. Let's denote the length of the spring when the 0.300 kg mass hangs from it as x₁ and the length of the spring when the 2.80 kg mass hangs from it as x₂.

From the problem, we have the following information:
x₁ = 0.190 m
x₂ = 0.750 m
m₁ = 0.300 kg
m₂ = 2.80 kg

To obtain the force constant, we can set up two equations using Hooke's Law for both situations:

For the first case when the 0.300 kg mass hangs:
F₁ = -kx₁

For the second case when the 2.80 kg mass hangs:
F₂ = -kx₂

Considering that the force applied to both cases is due to the weight of the masses, we can set up the following equations:

F₁ = m₁g
F₂ = m₂g

where g is the acceleration due to gravity (approximately 9.8 m/s²).

Substituting the force expressions into the Hooke's Law equations, we get:

m₁g = -kx₁
m₂g = -kx₂

Rearranging the equations to solve for k:

k = -m₁g/x₁
k = -m₂g/x₂

Now we can substitute the given values into the equations to find the force constant:

k = -(0.300 kg)(9.8 m/s²) / 0.190 m
k = -(2.80 kg)(9.8 m/s²) / 0.750 m

Evaluating these equations will give us the force constants for the spring.

(b) The unloaded length of the spring is the length of the spring when no mass is attached to it and there is no external force acting on it. This can be found by rearranging Hooke's Law:

F = -kx

When there is no force applied (F = 0), the equation becomes:

0 = -k * x_unloaded

Rearranging the equation to solve for the unloaded length:

x_unloaded = 0 / -k

Since the force constant k is nonzero, the unloaded length of the spring is 0 meters.

So, to summarize:

(a) The force constant of the spring can be found using the equations:
k = -(0.300 kg)(9.8 m/s²) / 0.190 m
k = -(2.80 kg)(9.8 m/s²) / 0.750 m

(b) The unloaded length of the spring is 0 meters.