A copper (shear modulus 4.2 x 1010 N/m2) cube, 0.324 m on a side, is subjected to two shearing forces, each of magnitude F = 6.93 x 10 5 N (see the drawing). Find the angle (in degrees), which is one measure of how the shape of the block has been altered by shear deformation.
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To find the angle by which the shape of the copper cube has been altered by shear deformation, we will use the formula:
θ = (F * L) / (G * A)
Where:
θ is the angle of shear deformation
F is the magnitude of the shearing force
L is the length of the cube
G is the shear modulus of copper
A is the area of the face being sheared
Given:
F = 6.93 x 10^5 N
L = 0.324 m
G = 4.2 x 10^10 N/m^2
The area of the face being sheared can be calculated as the length of the cube multiplied by the length of an edge side:
A = L^2 = (0.324 m)^2 = 0.104976 m^2
Now, we can substitute the given values into the formula to find the angle:
θ = (6.93 x 10^5 N * 0.324 m) / (4.2 x 10^10 N/m^2 * 0.104976 m^2)
Calculating this expression gives us:
θ ≈ 4.46 x 10^(-4) rad
To convert this angle to degrees, we can use the formula:
θ (degrees) = θ (radians) * (180 / π)
Substituting the value, we get:
θ (degrees) ≈ 4.46 x 10^(-4) rad * (180 / π)
Calculating this expression gives us:
θ (degrees) ≈ 0.0255 degrees
Therefore, the angle by which the shape of the copper cube has been altered by shear deformation is approximately 0.0255 degrees.
To find the angle of shear deformation, we need to use the shear modulus and the magnitude of the shearing forces. The formula for the angle of shear deformation is given by:
θ = (F * L) / (G * A)
where:
θ is the angle of shear deformation,
F is the magnitude of the shearing force,
L is the length of the side of the cube,
G is the shear modulus of the material, and
A is the area of the face of the cube on which the shearing forces are applied.
Given values:
Shear modulus (G) of copper = 4.2 x 10^10 N/m^2
Length of the side of the cube (L) = 0.324 m
Magnitude of the shearing forc