a cylinder of diameter 1cm at 30 degrees celcius is to be slid into a steel plate. the hole has a diameter 0.9997cm at 30 degrees celcius. to what temperature must the plate be heated? for steel
As I recall, we have the thermal expansion equation is
dL = Lα dT
where α for steel is 13.0*10^-6 m/m /°K
So, since we want the diameter of the hole to increase by 0.0003 cm, we want the circumference to increase by .0003pi = .0009425 cm = 9.425*10^-6 m
So, now we just have to plug those numbers into our equation:
9.425*10^-6 = (3.14065*10^-2)(13.0*10^-6) dT
dT = 23.08 °K
So, we have to raise the temperature by 23°, to 53°C
To determine the temperature to which the steel plate must be heated in order for the 1cm diameter cylinder to fit into the 0.9997cm diameter hole at 30 degrees Celsius, we can use the formula for thermal expansion.
The thermal expansion of a material, such as steel, can be calculated using the formula:
ΔL = α * L * ΔT
Where:
ΔL is the change in length or diameter,
α is the coefficient of linear expansion for the material,
L is the original length or diameter,
ΔT is the change in temperature.
For steel, the average coefficient of linear expansion is approximately 12 × 10^(-6) per degree Celsius.
In this case, the change in diameter of the hole is ΔD = 1cm - 0.9997cm = 0.0003cm.
Now, let's calculate the change in temperature (ΔT).
ΔL = α * L * ΔT
0.0003cm = (12 × 10^(-6) / °C) * 0.9997cm * ΔT
Solving for ΔT:
ΔT = 0.0003cm / (12 × 10^(-6) / °C) * 0.9997cm
ΔT ≈ 0.0125°C
Therefore, the steel plate must be heated by approximately 0.0125 degrees Celsius in order for the cylinder to fit into the hole.
To calculate the temperature to which the steel plate must be heated, we can use the concept of thermal expansion and the formula for linear expansion.
The linear expansion for a solid material can be given by the formula:
ΔL = α * L * ΔT
Where:
ΔL is the change in length
α is the coefficient of linear expansion
L is the original length of the material
ΔT is the change in temperature
Since we are dealing with a cylinder, the change in length can be approximated as the difference in radius.
Given:
Diameter of the cylinder at 30 degrees Celsius (original diameter) = 1 cm
Diameter of the hole in the steel plate at 30 degrees Celsius = 0.9997 cm
First, we need to calculate the original radius of the cylinder and the hole:
Original radius of the cylinder = Diameter of the cylinder / 2 = 1 cm / 2 = 0.5 cm
Original radius of the hole = Diameter of the hole / 2 = 0.9997 cm / 2 = 0.49985 cm
Now, we can calculate the change in radius:
Change in radius = Original radius of the cylinder - Original radius of the hole
= 0.5 cm - 0.49985 cm
≈ 0.00015 cm
Since we are given the original and final temperature as 30 degrees Celsius, ΔT = 0, and thus the change in radius is due to the change in temperature.
The coefficient of linear expansion for steel is approximately 12 × 10^-6 per degree Celsius.
Using the formula mentioned earlier, we can rearrange it to find the change in temperature:
ΔT = ΔL / (α * L)
Substituting the values, we have:
0.00015 cm = (0.000015 cm per degree Celsius) * (0.5 cm) * ΔT
Now, we can solve for ΔT:
ΔT = 0.00015 cm / (0.000015 cm per degree Celsius * 0.5 cm)
≈ 20 degrees Celsius
Therefore, the steel plate must be heated to approximately 20 degrees Celsius to accommodate the cylinder with a diameter of 1 cm at 30 degrees Celsius into the hole with a diameter of 0.9997 cm.