A thin spherical shell of silver has an inner radius of 8.74 x 10-2 m when the temperature is 29.4 °C. The shell is heated to 145 °C. Find the change in the interior volume of the shell.
To find the change in the interior volume of the shell, we need to calculate the initial and final volumes and then find the difference between them.
Step 1: Calculate the initial volume V1 of the shell at 29.4 °C.
The volume of a thin spherical shell is given by the formula:
V = (4/3)π(R^3 - r^3)
where R is the outer radius and r is the inner radius.
Given:
Inner radius at initial temperature (r1) = 8.74 x 10^-2 m
Since the shell is thin, the difference between the outer and inner radius should be small.
So, we can assume the outer radius (R1) at initial temperature is slightly larger than the inner radius (r1):
R1 ≈ r1
Using the formula for volume:
V1 = (4/3)π(R1^3 - r1^3) = (4/3)π(r1^3 - r1^3) = 0
Therefore, the initial volume V1 is zero.
Step 2: Calculate the final volume V2 of the shell at 145 °C.
Given:
Inner radius at final temperature (r2) = 8.74 x 10^-2 m
Temperature change (ΔT) = 145 °C - 29.4 °C = 115.6 °C
Linear coefficient of expansion (α) for silver = 19 x 10^-6 °C^-1
To find the change in radius (Δr), we use the formula:
Δr = α * r * ΔT
Substituting the given values:
Δr = (19 x 10^-6 °C^-1) * (8.74 x 10^-2 m) * (115.6 °C)
= 0.019 m
Therefore, the final radius (r2) is:
r2 = r1 + Δr
= 8.74 x 10^-2 m + 0.019 m
= 0.1094 m
Using the formula for volume:
V2 = (4/3)π(R2^3 - r2^3)
= (4/3)π(0.1094^3 - 8.74 x 10^-2^3)
≈ 6.466 x 10^-4 m^3
Step 3: Find the change in volume ΔV = V2 - V1.
ΔV = 6.466 x 10^-4 m^3 - 0
= 6.466 x 10^-4 m^3
Therefore, the change in interior volume of the shell is approximately 6.466 x 10^-4 m^3.
To find the change in the interior volume of the silver shell, we can start by calculating the initial and final interior volumes, and then finding the difference.
Step 1: Calculate the initial and final interior volumes
The initial interior volume (V1) of the shell can be calculated using the formula for the volume of a spherical shell:
V1 = (4/3) * π * ((r1 + t)^3 - r1^3)
where r1 is the initial inner radius of the shell (8.74 x 10^-2 m), and t is the thickness of the shell.
Given that this is a thin spherical shell, we can assume that the thickness of the shell is very small compared to the radius, so t can be neglected. Therefore, the formula simplifies to:
V1 = (4/3) * π * (r1^3)
Similarly, the final interior volume (V2) can be calculated using the final inner radius (r2) of the shell:
V2 = (4/3) * π * (r2^3)
Step 2: Calculate the change in interior volume
The change in interior volume (ΔV) is then given by the difference between the final and initial volumes:
ΔV = V2 - V1
Now we can substitute the values and calculate the answer.
Additional information needed: The question does not provide the final inner radius (r2) of the silver shell. Please provide this value in order to proceed with the calculations.