a spiral spring of material length 20cm has a scale pan hanging freely in its lower end. when an object of mass 40g is placed in the pan its length becomes 21.8cm. when the object is replaced with another of mass 60g the length becomes 22.5cm. calculate the mass of the scale pan.

Question

a spiral spring of material length 20cm has a scale pan hanging freely in its lower end. when an object of mass 40g is placed in the pan its length becomes 21.8cm. when the object is replaced with another of mass 60g the length becomes 22.5cm. calculate the mass of the scale pan.

Answer 1

To solve this problem, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to its change in length.

Let's denote the original length of the spring as L0, the mass of the object placed in the pan as m, and the change in length of the spring as ΔL.

From the given information, we have:

L0 = 20 cm
ΔL1 = 21.8 cm - 20 cm = 1.8 cm
ΔL2 = 22.5 cm - 20 cm = 2.5 cm
m1 = 40 g
m2 = 60 g

Using Hooke's Law, we can write the equation:

F = k * ΔL

Here, k represents the spring constant, which is a property of the spring material. Let's assume k remains constant in this problem.

To find the spring constant, we can rearrange the equation:

k = F / ΔL

Now, we need to find the force exerted by the spring for each object's mass.

For object 1 of mass m1 = 40 g:
F1 = k * ΔL1

For object 2 of mass m2 = 60 g:
F2 = k * ΔL2

Since the scale pan is hanging freely at the lower end of the spring, it doesn't affect the length of the spring. Therefore, the force exerted by the spring when only the scale pan is hanging will be zero.

Thus, the net force exerted by the spring when object 2 is in the pan is equal to the force exerted by the scale pan alone:

F2 = F_pan

Now, we can set up an equation to solve for the mass of the scale pan.

F2 = F_pan
k * ΔL2 = m_pan * g (where g is the acceleration due to gravity)

Rearranging this equation, we find:

m_pan = (k * ΔL2) / g

To solve for m_pan, we need to find the spring constant (k). We can use the force exerted by the spring when object 1 is in the pan to find k.

F1 = k * ΔL1

Rearranging this equation, we find:

k = F1 / ΔL1

Let's calculate the values:

k = F1 / ΔL1 = (m1 * g) / ΔL1 (where g is the acceleration due to gravity)

Then, calculate m_pan:

m_pan = (k * ΔL2) / g

Substituting the known values:

k = (m1 * g) / ΔL1 = (40 g * 9.8 m/s^2) / 1.8 cm
m_pan = (k * ΔL2) / g = [(40 g * 9.8 m/s^2) / 1.8 cm] * 2.5 cm

Performing the calculations correctly will give you the mass (in grams) of the scale pan.