A spiral spring of natural length of 20.0cm has a scale pan hanging freely on its lower end. When an object of mass 40g is placed in a pan. It length become 21.80cm when another object of mass 60g is placed in the pan the length becomes 22.05cm . Calculate the mass of the scale pan
so a 20g increase (40 to 60g) caused a deformtion of (22.05-21.8) 0.75cm
f=kx or x= f/x=20/.75=80/3=2.66g/cm
now originally,
f=kx
(40-Masspan)=2.66g/cm * 1.80cm
solve for mass pan
September 25 2020
To calculate the mass of the scale pan, we need to determine the change in length of the spiral spring caused by the mass of the objects.
Given:
Natural length of the spring (L₀) = 20.0 cm
Length when a 40g object is placed (L₁) = 21.80 cm
Length when a 60g object is placed (L₂) = 22.05 cm
Change in length caused by the 40g object (ΔL₁) = L₁ - L₀
Change in length caused by the 60g object (ΔL₂) = L₂ - L₀
First, let's calculate the change in length caused by the 40g object:
ΔL₁ = L₁ - L₀ = 21.80 cm - 20.0 cm
ΔL₁ = 1.80 cm
Next, let's calculate the change in length caused by the 60g object:
ΔL₂ = L₂ - L₀ = 22.05 cm - 20.0 cm
ΔL₂ = 2.05 cm
Now, we can use Hooke's Law to calculate the mass of the scale pan.
Hooke's Law states that the force exerted by a spring is proportional to its extension/displacement.
Therefore, we can set up the following equation:
F₁ = k * ΔL₁ (1)
F₂ = k * ΔL₂ (2)
Where F₁ and F₂ are the forces exerted by the spring due to the objects' masses, and k is the spring constant.
We can assume that the spring constant (k) remains constant for small displacements.
Since the scale pan's mass is being calculated, we can assume that the force F₁ caused by the 40g object equals the force F₂ caused by the 60g object.
Therefore, we can equate equations (1) and (2):
k * ΔL₁ = k * ΔL₂
Simplifying the equation:
ΔL₁ = ΔL₂
Substituting the values:
1.80 cm = 2.05 cm
As the above equation doesn't hold true, the assumption that F₁ = F₂ is incorrect.
Hence, we cannot calculate the exact mass of the scale pan with the given information.
To solve this problem, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. In this case, we need to find the force exerted by the spring when different masses are added to the scale pan.
Here's how you can calculate the mass of the scale pan:
Step 1: Determine the spring constant (k)
The spring constant (k) represents the stiffness of the spring and can be calculated using the formula:
k = (F/m) / x
Where:
F is the force applied to the spring,
m is the mass applied,
x is the change in length of the spring.
For the first scenario, when an object of mass 40g is placed in the pan, the change in length of the spring is:
Δx1 = 21.80 cm - 20.00 cm = 1.80 cm = 0.018 m
And the force applied to the spring:
F1 = mg = (0.04 kg) * (9.8 m/s²) = 0.392 N
Now we can calculate the spring constant (k) using the above formula:
k = (F1/m) / x1 = (0.392 N / 0.04 kg) / 0.018 m = 10.88 N/m
Step 2: Calculate the extension of the spring for the second scenario:
As another object of mass 60g is placed in the pan, the change in length of the spring is:
Δx2 = 22.05 cm - 20.00 cm = 2.05 cm = 0.0205 m
Step 3: Find the force exerted by the spring in the second scenario:
Using Hooke's Law (F = k * x), we can calculate the force exerted by the spring for the second scenario:
F2 = k * Δx2 = (10.88 N/m) * (0.0205 m) = 0.22264 N
Step 4: Calculate the total force exerted by the spring:
Since the scale pan is hanging freely, the force exerted by the spring should be equal to the weight of the pan and the masses it contains. Therefore:
F_total = F1 + F2 = 0.392 N + 0.22264 N = 0.61464 N
Step 5: Calculate the mass of the scale pan:
Using the formula F = mg, where F_total is the force exerted by the spring and g is the acceleration due to gravity (9.8 m/s²), we can find the mass (m) of the scale pan:
m = F_total / g = 0.61464 N / 9.8 m/s² ≈ 0.063 kg (or 63g)
Therefore, the mass of the scale pan is approximately 63g.