A brass lid screws tightly onto a glass jar at 20 oC. To help open the jar, it can be
placed into a bath of hot water. After this treatment, the temperature of the lid and the
jar are both 60 oC. The inside diameter of the lid is 8.0 cm at 20 oC. Find the size of the
gap (difference in radius) that develops by this procedure.
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To find the size of the gap (difference in radius) that develops when the jar is placed in hot water, we can use the principle of thermal expansion.
First, let's determine the original radius of the lid at 20°C and convert it to meters, as the SI unit for length is meters:
Radius (R) = 8.0 cm = 0.08 m
Next, we need to find the change in temperature (ΔT) experienced by the lid. The initial temperature is 20°C, and after being placed in the hot water bath, it increases to 60°C. Therefore:
ΔT = 60°C - 20°C = 40°C
Now, we need to find the coefficient of linear expansion (α) for brass. The coefficient of linear expansion represents the change in length per unit length per degree Celsius. For brass, the value of α is approximately 19 x 10^-6 °C^-1.
Using the formula for thermal expansion, we can calculate the change in radius (ΔR) using the equation:
ΔR = α * R * ΔT
Substituting the values:
ΔR = (19 x 10^-6 °C^-1) * (0.08 m) * (40°C)
Simplifying the calculation:
ΔR = 0.000076 m = 7.6 x 10^-5 m
Therefore, the size of the gap (difference in radius) that develops by placing the jar in hot water is approximately 7.6 x 10^-5 meters or 0.076 mm.