A spiral spring of natural 20.00cm has a scale pan hanging freely in it's lower end. When an object of mass 40g is placed in the pan,its length becomes 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?
recall that the displacement is proportional to the mass. So, if the scale pan has mass m,
k(m+40) = 1.80
k(m+60) = 2.05
Now just solve for m.
These are the equations if the two weighings are done separately. If the 60g mass is added to the 40g mass, the 2nd equation needs to change...
Well, it seems like that scale pan is really pulling the spring down! But don't worry, I'm here to help you solve this mathematical mystery.
Let's start by calculating the change in length when the 40g object is placed in the pan. We have:
ΔL1 = 21.80 cm - 20.00 cm
ΔL1 = 1.80 cm
Now, let's calculate the change in length when the 60g object is added:
ΔL2 = 22.05 cm - 20.00 cm
ΔL2 = 2.05 cm
We know that the extension of the spring is directly proportional to the mass added. So, we can set up a proportion:
ΔL1/40g = ΔL2/m
Substituting the values we have:
1.80 cm/40g = 2.05 cm/m
Now, let's cross-multiply and solve for m:
1.80 cm * m = 40g * 2.05 cm
m = (40g * 2.05 cm) / 1.80 cm
m ≈ 45.56 g
So, the mass of the scale pan is approximately 45.56 grams. It seems like the pan is a bit heavier than expected, doesn't it? Maybe it's been eating too much!
To calculate the mass of the scale pan, we need to determine the change in length of the spiral spring caused by the additional mass.
Step 1: Calculate the change in length for the first object.
Given:
Original length of the spring (Lo) = 20.00 cm
Length of the spring with the 40g object (L1) = 21.80 cm
Change in length for the first object (ΔL1) = L1 - Lo
ΔL1 = 21.80 cm - 20.00 cm
ΔL1 = 1.80 cm
Step 2: Calculate the change in length for the second object.
Given:
Original length of the spring (Lo) = 20.00 cm
Length of the spring with the 60g object (L2) = 22.05 cm
Change in length for the second object (ΔL2) = L2 - Lo
ΔL2 = 22.05 cm - 20.00 cm
ΔL2 = 2.05 cm
Step 3: Calculate the change in length per unit mass for both objects.
Change in length per unit mass for the first object (ΔL1/m1) = ΔL1 / m1
Where m1 is the mass of the first object (40g).
ΔL1/m1 = 1.80 cm / 40 g
ΔL1/m1 = 0.045 cm/g
Change in length per unit mass for the second object (ΔL2/m2) = ΔL2 / m2
Where m2 is the mass of the second object (60g).
ΔL2/m2 = 2.05 cm / 60 g
ΔL2/m2 = 0.034 cm/g
Step 4: Calculate the average change in length per unit mass.
Average change in length per unit mass (ΔL_avg/m_avg) = (ΔL1/m1 + ΔL2/m2) / 2
ΔL_avg/m_avg = (0.045 cm/g + 0.034 cm/g) / 2
ΔL_avg/m_avg = 0.079 cm/g / 2
ΔL_avg/m_avg = 0.0395 cm/g
Step 5: Calculate the mass of the scale pan.
Given that the scale pan has no mass (m_scale_pan = 0g).
ΔL_avg/m_avg = ΔL_scale_pan / m_scale_pan
0.0395 cm/g = 0.025 cm / m_scale_pan
To find m_scale_pan, we rearrange the equation:
m_scale_pan = (0.025 cm) / (0.0395 cm/g)
m_scale_pan = 0.6329 g
Therefore, the mass of the scale pan is approximately 0.6329 grams.
To calculate the mass of the scale pan, we need to use Hooke's Law, which relates the force applied to an object to the extension or compression of a spring. The formula for Hooke's Law is:
F = kx
Where:
F is the force applied to the spring
k is the spring constant
x is the change in length of the spring
In this case, the change in length of the spring is given by:
Δx = final length - initial length
First, we need to calculate the spring constant (k) of the spiral spring. For this, we need two sets of data.
Let's consider the first set of data:
The initial length (x1) is 20.00 cm, and when a mass of 40 g is placed in the pan, the final length (x2) becomes 21.80 cm.
Δx1 = x2 - x1
= 21.80 cm - 20.00 cm
= 1.80 cm
Now, let's calculate the force (F1) applied to the spring for the first set of data:
F1 = k * Δx1
For the second set of data:
The initial length (x1) is 20.00 cm, and when a mass of 60 g is placed in the pan, the final length (x2) becomes 22.05 cm.
Δx2 = x2 - x1
= 22.05 cm - 20.00 cm
= 2.05 cm
Now, let's calculate the force (F2) applied to the spring for the second set of data:
F2 = k * Δx2
Since the mass of the object is given in grams, we need to convert it to kilograms to calculate the force. 1 gram is equal to 0.001 kg.
Now, we can set up two equations using the forces calculated above:
F1 = m1 * g (1)
F2 = m2 * g (2)
where m1 is the mass of the 40 g object and m2 is the mass of the 60 g object, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the forces, we have:
k * Δx1 = m1 * g (3)
k * Δx2 = m2 * g (4)
Let's solve equations (3) and (4) simultaneously to find the values of m1 and m2.