To make a secure fit, rivets that are larger than the rivet hole are often used and the rivet is cooled (usually in dry ice) before it is placed in the hole. A steel rivet 1.873 cm in diameter is to be placed in a hole 1.871 cm in diameter in a metal at 20°C. To what temperature must the rivet be cooled if it is to fit in the hole?
To determine the temperature to which the rivet must be cooled in order to fit in the hole, we can use the principle of thermal expansion. The basic idea is that when a material is heated, it expands, and when it is cooled, it contracts.
In this case, we need to shrink the diameter of the rivet slightly so that it can fit inside the hole. We know the initial diameter of the rivet is 1.873 cm, and the diameter of the hole is 1.871 cm. We need to find the temperature at which the rivet's diameter will contract by 0.002 cm, so it will match the hole's diameter.
The formula for thermal expansion is:
ΔL = α * L * ΔT
Where:
ΔL is the change in length or diameter
α is the coefficient of linear expansion
L is the original length or diameter
ΔT is the change in temperature
We know ΔL = 0.002 cm, L = 1.873 cm, and α for steel is about 0.000012 per °C.
Substituting these values into the formula, we can solve for ΔT:
0.002 cm = (0.000012/°C) * (1.873 cm) * ΔT
Simplifying the equation gives:
ΔT = 0.002 cm / (0.000012/°C * 1.873 cm)
Calculating this expression gives:
ΔT ≈ 139°C
Therefore, the rivet must be cooled to approximately -139°C to fit in the hole at 20°C.
To determine the temperature to which the rivet must be cooled, we can use the principle of thermal expansion. The idea is to find the temperature at which the rivet's diameter matches the hole's diameter, considering the expansion and contraction of both the rivet and the hole.
Here's how you can calculate it step by step:
1. Find the change in diameter of the rivet. We have:
ΔD = D_rivet - D_hole
ΔD = 1.873 cm - 1.871 cm
ΔD = 0.002 cm
2. Determine the linear expansion coefficient (α) of the metal. Since we are given the material as steel, we can look up or assume a value for steel. Let's assume α = 12 x 10^-6 °C^-1 for steel.
3. Use the formula for linear thermal expansion:
ΔL = α * L * ΔT
Where ΔL is the change in length, α is the linear expansion coefficient, L is the original length, and ΔT is the change in temperature.
Since we are dealing with diameter, and diameter (D) is two times the length (L), we can rewrite it as:
ΔD = 2 * α * D * ΔT
Rearranging the equation to solve for ΔT:
ΔT = ΔD / (2 * α * D)
ΔT = 0.002 cm / (2 * 12 x 10^-6 °C^-1 * 1.873 cm)
4. Calculate ΔT:
ΔT = 0.002 cm / (2 * 12 x 10^-6 °C^-1 * 1.873 cm)
ΔT ≈ 56.72°C
Thus, the rivet needs to be cooled to approximately 56.72°C in order to fit in the 1.871 cm diameter hole at 20°C.
so helpfull
First, look up the coefficient of thermal expansion of steel. Call it "a". Let the change in temperature necessary to shrink the rivet to fit the hole be "delta T". It will be a negative number in this case. It depends up the type of steel you have: structural (carbon) steel or stainless steel. They should have told you which it is. A reasonable number to use is
a = 13*10^-6 (deg C)^-1
D2 = final rivet diameter
D1 = initial rivet diameter
Solve the equation
D2 - D1 = -0.002 = 1.873*a*(delta T)
delta T = -0.00107/a
and add that to the initial temperature. Since deelta T is negative, you will end up with a colder temperature