At 20.0 C, an alumium ring has an inner diameter of 5.0000 cm and a brass rod has a diameter of 5.0500 cm.
a. If only the ring is heated, what temperature must it reach so that it will just slip over the rod?
b. WHAT IF? If both are heated together, what temperature must they both reach so that the ring just slips over the rod? Would the latter process work?
sorry., this one i really couldn't figure out. although the last 2 i got it after awhile.
calculate the circumference of the rod.
Calculate the difference in the circumerance of the ring and the rod.
a) LEt that be delta L
deltaL= Ringcirc*coeffAl*deltaT
solve for delta T.
b) for each, the original material Circumer + delta L is equal to the sum of L and delta L for the other.
aluminumcirc=rodcirc
but the rod expands, it has a delta V. So the delta L for it has to be determined by changes in V.
Volume= 4/3 PI r^3
dV= 4 PI r^2 dr= V(coefficientvolume)deltaT
or
dr= V*coeffV*deltaTemp/4PIr^2 where r is the original r. And
newCircumference= 2PI (r + dr)
Now, set the aluminum length heated equal to the brass change in circumference.
Solve for deltaTemp.
so i figured out the first part
but im still lost about part b
.
To calculate the temperature at which both the aluminum ring and the brass rod will just slip over each other when heated together, we need to find the difference in their circumferences after expansion.
Let's go step-by-step to solve part b:
1. Calculate the circumference of the rod:
Circumference = π * diameter
Circumference of the brass rod = π * 5.0500 cm
2. Calculate the change in circumference for the rod:
ΔCircumference_rod = New_Circumference - Original_Circumference
ΔCircumference_rod = New_Circumference - π * 5.0500 cm
3. Calculate the change in circumference for the aluminum ring due to heating:
ΔCircumference_ring = Ring_Circumference * coefficient_of_linear_expansion_aluminum * ΔTemperature
ΔCircumference_ring = 2π * (Inner_Diameter + ΔDiameter_ring) * coefficient_of_linear_expansion_aluminum * ΔTemperature
4. Set the change in circumference for both (ring and rod) equal to each other:
ΔCircumference_rod = ΔCircumference_ring
5. Substitute the expressions for ΔCircumference_rod and ΔCircumference_ring from steps 2 and 3:
π * 5.0500 cm - π * 5.0500 cm = 2π * (5.0000 cm + ΔDiameter_ring) * coefficient_of_linear_expansion_aluminum * ΔTemperature
6. Simplify the equation:
0 = 2 * (5.0000 cm + ΔDiameter_ring) * coefficient_of_linear_expansion_aluminum * ΔTemperature
7. Solve for ΔTemperature:
ΔTemperature = 0 (Since all the terms on the right-hand side of the equation are already determined)
This means that, in order for the ring to just slip over the rod when both are heated together, the temperature difference (ΔTemperature) should be 0. This implies that they will not slip over each other as they expand uniformly.
Therefore, the latter process would not work for the ring to slip over the rod when heated together.
To calculate the temperature at which both the ring and the rod will just slip over each other when heated together, you need to compare the changes in circumference of both objects.
Here's how you can approach part b):
1. Calculate the initial circumference of the rod:
- The formula for the circumference of a circle is C = 2πr, where r is the radius.
- Since the diameter of the rod is given as 5.0500 cm, the radius would be half of that, so r = 5.0500 cm / 2 = 2.5250 cm.
- Now use the formula to calculate the initial circumference: C_rod = 2π(2.5250 cm).
2. Calculate the change in circumference for the rod:
- As the rod is heated, it will expand, resulting in a change in circumference. This change is related to the change in volume.
- The formula to calculate the change in volume of a solid is ΔV = V * β * ΔT, where ΔV is the change in volume, V is the initial volume of the object, β is the coefficient of volume expansion, and ΔT is the change in temperature.
- Note that the ΔT in this case represents the temperature change from the initial temperature to the required temperature where the ring just slips over the rod.
- The initial volume of the rod can be calculated using V = (4/3)πr^3.
- Plug in the appropriate values into the equation and solve for ΔV.
- Now, use ΔV to find the change in radius (Δr) using the formula Δr = ΔV / (4πr^2).
- Finally, calculate the new circumference of the rod after expansion: C_rod_new = 2π(r + Δr).
3. Equate the circumference of the rod and the circumference of the expanded ring:
- Set the new circumference of the rod (C_rod_new) equal to the circumference of the expanded ring (C_ring + ΔL), where ΔL is the change in length of the ring due to heating.
- Solve this equation for ΔT.
By following these steps, you should be able to find the required temperature (ΔT) at which the ring will just slip over the rod when they are heated together.