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 that develops between the lid and the jar after being placed in hot water, we can use the principle of thermal expansion.
1. Firstly, we need to find the change in temperature (ΔT) between 20 oC and 60 oC:
ΔT = 60 oC - 20 oC
ΔT = 40 oC
2. Next, we need to determine the coefficient of linear expansion (α) for brass. The coefficient of linear expansion is a material-specific constant that tells us how much a material expands or contracts per degree Celsius. For brass, the coefficient of linear expansion is approximately 19 x 10^(-6) oC^(-1).
3. Now, we can calculate the change in radius (Δr) of the lid due to the change in temperature:
Δr = α * r * ΔT
Δr = (19 x 10^(-6) oC^(-1)) * (0.08 m) * (40 oC)
Δr = 0.0608 mm
Therefore, the size of the gap (difference in radius) that develops by placing the jar and lid in hot water is approximately 0.0608 mm.
To find the size of the gap that develops due to the temperature change, we need to take into account the thermal expansion of both the brass lid and the glass jar.
To solve this problem, we can use the formula for linear thermal expansion:
ΔL = α * L * ΔT,
where ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature.
In this case, since we are dealing with radii and diameter, we need to use the formula for radial or area thermal expansion, given by:
ΔA = 2πr Δr,
where ΔA is the change in area, r is the original radius, and Δr is the change in radius.
Now let's go step by step:
1. Convert the Celsius temperatures to Kelvin:
T1 = 20 + 273 = 293 K
T2 = 60 + 273 = 333 K
2. Find the change in radius for the brass lid:
We can use the formula for linear thermal expansion. The coefficient of linear expansion for brass is typically α = 19 × 10^-6 K^-1.
Δr_brass = α_brass * r_brass * ΔT = 19 × 10^-6 K^-1 * 8.0 cm * (333 K - 293 K) = 0.00672 cm
3. Find the change in radius for the glass jar:
The coefficient of linear expansion for glass is typically α = 8 × 10^-6 K^-1.
Δr_glass = α_glass * r_glass * ΔT = 8 × 10^-6 K^-1 * 8.0 cm * (333 K - 293 K) = 0.00288 cm
4. Calculate the net change in radius (gap):
Δr_net = Δr_brass - Δr_glass = 0.00672 cm - 0.00288 cm = 0.00384 cm
Therefore, the size of the gap (difference in radius) that develops by this procedure is approximately 0.00384 cm.
the coefficent of area expansion is twice that of the coeff of linear expansiosn.
http://www.engineeringtoolbox.com/linear-expansion-coefficients-d_95.html
From the increase in area, calculate the change in diameter.