A spherical steel ball has a diameter of 2.540cm at 25 degree Celsius. (a)What is its diameter when the temperature is raised to 100 degree Celsius? (b)What temperature change is required to increase its volume by 1%?
To solve these problems, we can use the concept of thermal expansion of materials. The linear thermal expansion coefficient (α) is a measure of how much a material expands or contracts with a change in temperature.
(a) To find the diameter of the steel ball when the temperature is raised to 100 degrees Celsius, we can use the formula for linear expansion:
ΔL = α * L0 * ΔT
where ΔL is the change in length, α is the linear expansion coefficient, L0 is the initial length, and ΔT is the change in temperature.
Since we are dealing with the diameter, which is twice the length, we can substitute 2L0 for ΔL in the formula:
ΔL = 2 * α * L0 * ΔT
Let's calculate it step by step:
1. Convert the diameter from centimeters to meters:
Initial diameter (D0) = 2.540 cm = 0.02540 m
2. Find the initial radius (R0):
Initial radius (R0) = D0 / 2 = 0.02540 m / 2 = 0.01270 m
3. Find the initial volume (V0) of the sphere:
Initial volume (V0) = (4/3) * π * R0^3
4. Find the change in temperature (ΔT):
ΔT = 100°C - 25°C = 75°C
5. Find the linear expansion coefficient for steel:
The linear expansion coefficient α for steel is approximately 12 x 10^-6 (1/°C) or 12 x 10^-6 m/m°C.
6. Calculate the change in diameter (ΔD):
ΔD = 2 * α * R0 * ΔT
Plug in the values:
ΔD = 2 * (12 x 10^-6) * (0.01270) * (75)
7. Calculate the new diameter (D):
D = D0 + ΔD
Plug in the values:
D = 0.02540 + ΔD
Simplify the equation and solve for D.
(b) To find the temperature change required to increase the volume by 1%, we can use the formula for the volume expansion:
ΔV = β * V0 * ΔT
where ΔV is the change in volume, β is the volume expansion coefficient, V0 is the initial volume, and ΔT is the change in temperature.
We are given that the change in volume is 1% of the initial volume. So:
ΔV = 0.01 * V0
Let's calculate it step by step:
1. Find the initial volume (V0) using the formula from above:
V0 = (4/3) * π * R0^3
2. Find the volume expansion coefficient (β) for steel:
The volume expansion coefficient (β) for steel is approximately 3α, where α is the linear expansion coefficient.
So, β = 3 * 12 x 10^-6 = 36 x 10^-6 (1/°C).
3. Substitute the given values into the formula:
0.01 * V0 = β * V0 * ΔT
Simplify the equation and solve for ΔT.