A lead sphere has a diameter that is 0.050% larger than the inner diameter of a steel ring when each has a temperature of 27.9 °C. Thus, the ring will not slip over the sphere. At what common temperature will the ring just slip over the sphere
To find the common temperature at which the ring will just slip over the sphere, we need to use the coefficient of linear expansion for both lead and steel.
The formula for the change in diameter of a material due to temperature change is given by:
ΔD = αDΔT
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
We know that the lead sphere's diameter is 0.050% larger than the inner diameter of the steel ring. Let's call the original diameter of the steel ring D and the original diameter of the lead sphere d.
Since the lead sphere's diameter is larger, we can express it as:
d = D + (0.050/100)D
= D(1 + 0.050/100)
Now, we need to find the change in temperature (ΔT) that will cause the ring to just slip over the sphere. When the ring is at its original temperature, it does not slip over the sphere, so their diameters must be the same:
D = d
Let's assume the final temperature at which the ring just slips over the sphere is T common to both materials.
Using the formula above, we can calculate the change in diameter for both materials:
For the steel ring:
ΔDsteel = αsteel*DΔT
For the lead sphere:
ΔDlead = αlead*dΔT
Since ΔDsteel = -ΔDlead (the steel ring shrinks while the lead sphere expands), we can set up the equation:
-αsteel*DΔT = αlead*dΔT
Now let's plug in the values we know:
-αsteel*D(T - 27.9) = αlead*d(T - 27.9)
We can simplify this equation by dividing both sides by ΔT and rearranging:
-αsteel*D = αlead*d
αsteel/αlead = d/D
Since we know the relationship between d and D, we can substitute in the expression for d/D:
αsteel/αlead = (1 + 0.050/100)
Now we can solve for T:
T - 27.9 = αsteel/αlead * 27.9
T = 27.9 + (αsteel/αlead) * 27.9
Therefore, the common temperature at which the ring will just slip over the sphere is 27.9 °C plus the product of the ratio of the steel's coefficient of linear expansion to the lead's coefficient of linear expansion and 27.9 °C.