A copper cube, 0.31 m on a side, is subjected to two shearing forces, each which has a magnitude F = 6.40 106 N (see the drawing). Find the angle θ (in degrees), which is one measure of how the shape of the block has been altered by shear deformation.
To find the angle θ, we need to use the concept of shear strain. Shear strain is defined as the change in shape of a material due to shearing forces.
The formula for shear strain is given by:
Shear Strain (γ) = Δx / h
Where:
Δx = displacement in the x-direction (horizontal)
h = height of the cube (0.31 m)
In this case, the displacement Δx is equal to half the length of the side of the cube (0.31 m / 2 = 0.155 m) since the forces are applied symmetrically.
Now that we have the shear strain (γ), we can relate it to the angle of deformation using the tangent function:
tan(θ) = γ
Rearranging the formula, we get:
θ = arctan(γ)
Let's calculate:
First, we need to find the shear strain (γ):
γ = Δx / h = 0.155 m / 0.31 m = 0.5
Now, we can find the angle (θ):
θ = arctan(0.5)
Calculating this value using a calculator or trigonometric table, we find that θ is approximately 26.57 degrees.
Therefore, the angle θ, which is one measure of how the shape of the block has been altered by shear deformation, is approximately 26.57 degrees.