How do i get the answer
A steel ball of diameter Lb = 4.196 cm is placed on a hole of diameter Lh = 4.042 cm on an aluminium plate. The initial temperature is 14 ¢XC. What would be the minimum temperature so that the ball will fit through the hole if both are heated? The linear expansion coefficients for steel and aluminium are £\steel = 1.100 ¡Ñ 10¡V5 K¡V1 and £\alu = 2.200 ¡Ñ 10¡V5 K¡V1, respectively. ans 3610 degrees celcius
First let's use symbols people can read.
a1 = 1.1*10^-5 C^-1 (thermal expansion cefficient of steel)
a2 = 2.2*10^-5 C^-1 (thermal expnsion cefficient of aluminum)
The hole and the ball will both expand in proportion to the linear expansion coefficient and the temperature rise, dT. The diameters will be equal when
4.042(1 + a2*dT) = 4.196(1 + a1*dT)
0.154 = (8.892 - 4.616)*10^-5*dT
= 4.276*10^-5*dT
dT = 3602 C
Add that to the initial temperature of 14 C and you get 3616 C
To determine the minimum temperature needed for the steel ball to fit through the hole, we need to consider the thermal expansion of both the steel ball and the aluminum plate.
The formula for thermal expansion is given by:
ΔL = αLΔT
Where:
ΔL is the change in length
α is the linear expansion coefficient
L is the original length
ΔT is the change in temperature
In this case, we need to find the change in diameter of both the steel ball and the hole. Since the diameter is twice the length, the formula becomes:
ΔD = 2αDLΔT
To simplify the equation, let's assume that the original diameter of the steel ball (D_st) is equal to the original diameter of the hole (D_h). Therefore, ΔD = 0 in this case.
Now, let's solve for ΔT, the change in temperature required for the steel ball to fit through the hole.
1. Substitute the known values:
ΔD = 0
α_st = 1.100 × 10^(-5) K^(-1) (linear expansion coefficient for steel)
α_alu = 2.200 × 10^(-5) K^(-1) (linear expansion coefficient for aluminum)
D = 4.196 cm (diameter of the steel ball)
2. Rearrange the equation to solve for ΔT:
0 = 2α_st D_st ΔT - 2α_alu D_alu ΔT
Since D_st = D_alu, we can simplify it as:
0 = 2(α_st - α_alu) D ΔT
3. Solve for ΔT:
ΔT = 0 / (2(α_st - α_alu) D)
= 0
This means that the change in temperature required for the steel ball to fit through the hole is 0 degrees Celsius. Therefore, the minimum temperature needed would be the initial temperature of 14 ¢XC.
Please note that there may be other factors to consider in real-world applications, such as additional clearance or tolerances. The provided answer assumes ideal conditions without any other constraints.