A spring is hung from a ceiling, and an object attached to its lower end stretches the spring by a distance d = 7.70 cm from its unstretched position when the system is in equilibrium as in the figure below. If the spring constant is 48.0 N/m, determine the mass of the object.

__________ kg

Wo = 48N/m * 0.077m = 3.696 N. = Wt. of

object.

m*g = 3.696 N.
m = 3.696/g = 3.696/9.8 = 0.377 kg.

Well, this situation seems quite "springtacular"! Let's dive into the calculations, shall we?

To find the mass of the object, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring.

In this case, the displacement (d) is given as 7.70 cm, which we should convert to meters because the spring constant is given in N/m. So, d = 0.077 m.

According to Hooke's Law, the force (F) exerted by the spring is given by F = k * d, where k is the spring constant.

Plugging in the given values, we have F = 48.0 N/m * 0.077 m.

Now let's introduce a little bit of Newton's second law, which states that force (F) is equal to mass (m) times acceleration (a).

Since the system is in equilibrium, the acceleration is zero. Therefore, we have F = m * a = m * 0, which means there is no net force acting on the object.

Equating these two expressions for F, we have 48.0 N/m * 0.077 m = m * 0, which simplifies to 3.696 N = 0.

Oh no! Something went wrong with our calculations! It seems like there has been a "mass-ive" mistake!

Considering that Newton's second law applies, we can rearrange the equation to solve for mass: m = F / a. Since the acceleration is zero, we can conclude that the mass of the object is also zero!

However, in reality, that can't be the case. So, let's double-check our calculations.

Actually, it looks like we made a mistake earlier. We need to consider the force exerted by the spring equals the weight of the object when in equilibrium.

The weight of an object is given by W = m * g, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s²).

So we can say that F = W = m * g.

Plugging in the value of F, we have 3.696 N = m * 9.8 m/s².

Dividing both sides of the equation by 9.8 m/s², we find m = 3.696 N / 9.8 m/s².

Approximating this value, we get m ≈ 0.3771 kg.

So, after all the "springing" around, the mass of the object is approximately 0.3771 kg.

To find the mass of the object, 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.

The formula for Hooke's Law is:

F = k * x

Where:
F is the force exerted by the spring (in Newtons),
k is the spring constant (in N/m),
and x is the displacement from the equilibrium position (in meters).

In this case, the displacement is given as d = 7.70 cm = 0.0770 m, and the spring constant is given as k = 48.0 N/m.

Substituting these values into the formula, we have:

F = (48.0 N/m) * (0.0770 m)
F = 3.696 N

The force exerted by the spring is 3.696 N.

Now, we can use Newton's second law, which states that the force acting on an object is equal to its mass multiplied by its acceleration.

The formula for Newton's second law is:

F = m * a

Where:
F is the force acting on the object (in Newtons),
m is the mass of the object (in kilograms),
and a is the acceleration of the object (in meters per second squared).

In this case, the force acting on the object is 3.696 N, and we can assume that the acceleration is zero since the object is in equilibrium.

Substituting these values into the formula, we have:

3.696 N = m * 0

Therefore, the mass of the object can be any value since the acceleration is zero.

So, the mass of the object is indeterminate.

To determine the mass of the object attached to the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement or stretch of the spring.

The formula for Hooke's Law is:

F = k * x

Where:
F is the force exerted by the spring (in newtons),
k is the spring constant (in newtons per meter),
x is the displacement or stretch of the spring (in meters).

In this case, the displacement or stretch of the spring is given as d = 7.70 cm = 0.077 m, and the spring constant is given as k = 48.0 N/m.

Plugging the given values into the Hooke's Law formula, we get:

F = (48.0 N/m) * (0.077 m)

Simplifying, we find the force exerted by the spring:

F = 3.696 N

Now, we can use Newton's Second Law of Motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration:

F = m * g

Where:
F is the force acting on the object (in newtons),
m is the mass of the object (in kilograms),
g is the acceleration due to gravity (approximately 9.8 m/s^2).

In this case, the force acting on the object is the force exerted by the spring (F = 3.696 N).

Plugging the given values into Newton's Second Law, we can solve for the mass of the object:

3.696 N = m * 9.8 m/s^2

Simplifying, we find the mass of the object:

m = 3.696 N / 9.8 m/s^2 ≈ 0.377 kg

Therefore, the mass of the object attached to the spring is approximately 0.377 kg.