A spherical steel ball bearing has a diameter of 2.540 cm at 23°C.
(a) What is the diameter when its temperature is raised to 106°C?
(b) What temperature change is required to increase its volume by 0.8%?
To find the diameter of the steel ball bearing when its temperature is raised to 106°C, and to determine the temperature change required to increase its volume by 0.8%, we can use the principles of thermal expansion and the formula for volume change due to temperature change.
(a) To find the diameter at a higher temperature, we need to consider the linear thermal expansion of the steel ball bearing. The formula for linear thermal expansion is given as:
ΔL = α * L * ΔT
Where:
ΔL is the change in length/size
α is the coefficient of linear expansion
L is the original length/size
ΔT is the change in temperature
In this case, we are interested in the change in diameter, which is twice the change in length of the radius. So, we can write:
ΔD = 2 * α * D * ΔT
Where:
ΔD is the change in diameter
α is the coefficient of linear expansion
D is the original diameter
ΔT is the change in temperature
The coefficient of linear expansion for steel can be found through reference sources. Let's assume α = 12 * 10^(-6) per °C for steel.
First, we need to calculate the change in temperature:
ΔT = (final temperature) - (initial temperature)
ΔT = 106°C - 23°C
ΔT = 83°C
Now we can calculate the change in diameter:
ΔD = 2 * (12 * 10^(-6) per °C) * (2.540 cm) * (83°C)
Using this formula, you can calculate the change in diameter when the temperature is raised to 106°C.
(b) To find the temperature change required to increase the volume by 0.8%, we can use the formula for volume expansion due to temperature change:
ΔV = β * V * ΔT
Where:
ΔV is the change in volume
β is the coefficient of volume expansion
V is the original volume
ΔT is the change in temperature
In this case, we know that ΔV/V = 0.008 (0.8%), so:
0.008 = β * ΔT
Now we need to find the coefficient of volume expansion for steel, which can also be found through reference sources. Let's assume β = 36 * 10^(-6) per °C for steel.
Then we can rearrange the equation to solve for ΔT:
ΔT = 0.008 / (36 * 10^(-6) per °C)
Using this formula, you can calculate the temperature change required to increase the volume by 0.8%.