A 5cm cube of steel is subjected to a uniform pressure of 2,100 Kg/cm2
acting on all faces. Determine the change in dimension between two
parallel faces of the cube. E = 21 x 105 kg/cm2 and μ = ¼.
Well, if I were that steel cube, I'd definitely be feeling pressured right now! But let's calculate the change in dimension with some number crunching.
First, we need to find the stress on the cube. Stress is the pressure applied per unit area, so the stress on each face would be 2,100 Kg/cm2.
Now, let's find the strain. Strain is the change in dimension divided by the original dimension. Since the applied pressure is uniform on all faces, the original dimension of each face is 5 cm.
Now, let's use Hooke's Law to find the strain. Hooke's Law states that stress is equal to the product of Young's modulus and strain. The formula for strain is: strain = (stress / (E * (1 - μ))).
Plugging in the values, we have: strain = (2,100 Kg/cm2) / (21 x 105 kg/cm2 * (1 - 1/4)).
Simplifying the equation, we have: strain = (2,100) / (21 * 4/3).
So, the strain is: strain = 100.
Now, let's calculate the change in dimension. The change in dimension is equal to the original dimension multiplied by the strain. The original dimension is 5 cm, so the change in dimension would be: change in dimension = 5 cm * 100.
Therefore, the change in dimension between two parallel faces of the steel cube is 500 cm. That's quite a stretch!
To determine the change in dimensions between two parallel faces of the cube, we can use the formula for the change in volume due to applied pressure:
ΔV = V * ΔP / K
Where:
ΔV = Change in volume
V = Initial volume
ΔP = Change in pressure
K = Bulk modulus of the material
Step 1: Calculate the initial volume of the cube
The initial volume of the cube can be calculated by taking the cube of its side length:
V = (side length)^3 = (5 cm)^3 = 125 cm³
Step 2: Calculate the change in pressure
The change in pressure can be calculated by subtracting the initial pressure from the final pressure:
ΔP = 2,100 Kg/cm²
Step 3: Calculate the bulk modulus
The bulk modulus, K, can be calculated using the formula:
K = E / (3(1-2μ))
Given that E = 21 x 10^5 kg/cm² and μ = 1/4:
K = (21 x 10^5) / (3(1-2(1/4)))
= (21 x 10^5) / (3(1-1/2))
= (21 x 10^5) / (3(1/2))
= (21 x 10^5) / (3/2)
= (21 x 10^5) x (2/3)
= 14 x 10^5 kg/cm²
Step 4: Calculate the change in volume
Now we can substitute the values into the formula:
ΔV = V * ΔP / K
ΔV = 125 cm³ * 2,100 Kg/cm² / (14 x 10^5 kg/cm²)
Simplifying:
ΔV = 9 cm³
Step 5: Calculate the change in dimension
The change in dimension, ΔL, can be calculated by dividing the change in volume by the original cross-sectional area:
ΔL = ΔV / (cross-sectional area)
Since it is a cube, all faces are identical, and the cross-sectional area can be calculated by taking the square of the side length:
cross-sectional area = (side length)^2
cross-sectional area = (5 cm)^2 = 25 cm²
Now we can substitute the values into the formula:
ΔL = 9 cm³ / 25 cm²
Simplifying:
ΔL = 0.36 cm
Therefore, the change in dimension between two parallel faces of the cube is 0.36 cm.
To determine the change in dimension between two parallel faces of the steel cube, we can use the formula for bulk modulus and Poisson's ratio.
The formula for bulk modulus (K) is:
K = (E) / (3(1 - 2μ))
Where:
E = Young's modulus
μ = Poisson's ratio
Given that E = 21 x 10^5 kg/cm^2 and μ = 1/4, we can substitute the values:
K = (21 x 10^5 kg/cm^2) / (3(1 - 2(1/4)))
Simplifying further:
K = (21 x 10^5 kg/cm^2) / (3(1 - 1/2))
K = (21 x 10^5 kg/cm^2) / (3(1/2))
K = (21 x 10^5 kg/cm^2) / (3/2)
K = (21 x 10^5 kg/cm^2) / (1.5)
K = 14 x 10^5 kg/cm^2
Now, we can use the formula for the change in dimension:
ΔL/L = -P / K
Where:
ΔL = Change in dimension
L = Original length
P = Uniform pressure applied
Given that the uniform pressure (P) is 2,100 kg/cm^2 and the original length (L) is 5 cm, we can substitute the values:
ΔL/L = -2,100 kg/cm^2 / 14 x 10^5 kg/cm^2
ΔL/L = -15 x 10^-3
Finally, we can calculate the change in dimension between two parallel faces:
ΔL = (-15 x 10^-3) x 5 cm
ΔL = -0.075 cm (the negative sign indicates a decrease in dimension)
Therefore, the change in dimension between two parallel faces of the steel cube is -0.075 cm.