A square rod of length L0 = 2.45m and sidelength d = 6.78cm is compressed with...

A square rod of length L0 = 2.45m and sidelength d = 6.78cm is compressed with a force of |Fc| = 90.8N in the direction of the long axis. (a) What is the magnitude of the scalar tensile stress? (b) If we assume that the area of the square rod is approximately constant and its final resting length is L = 2.35m, then what is the Young's modulus of the square rod? Now assume that the rod is stuck between two slabs that are separated by the sidelength of the rod. (c) What is the shear stress when the magnitude of the force of the top slab on the square rod in the direction of the long axis of the rod is |Fs| = 73.8N? (d) If the total distance that the top of the rod moved, as compared to the bottom of the rod, is ?x = 0.206cm, then what is the shear modulus of the rod in this geometry?

Solutions

Expert Solution

Starting with all the given values,

(A) Since the force applied is in the direction of the long axis, the only way to measure tensile stress would be to apply it vertically (i.e on the small square).Tensile stress (or compressive stress since the force here is compressing the solid rod) is given by,

(B) To calculate Young's Modulus, we need to calculate the strain induced due to the force applies. Strain is defined as the ratio of change in length of an object in the direction of the force applied to the original length of the object. So, the strain is,

Young's modulus is an important property of a solid and is given by,

(C) The rod is now placed between two slabs separated by the sidelength of the rod. So, the forces will act along the length on the rod (i.e. the rectangle part of the rod). Force is applied again, in the direction of the long axis of the rod. Now, the equivalent stress is called shear stress. Imagine pushing the slab across the length of a surface, like you would with a sand paper, (that's the direction of the force here), albeit with much much higher force. It is still stress and therfore will be measured Pascals.

(D) Now the shear strain is a bit different from normal strain. If let's say the force applied is on the top, then the ratio of displacement of the top part of the rod (in the direction parallel to the force) with respect to the bottom part to the length transverse to it is called shear strain.

Similar to the Young's Modulus, Shear Modulus is given by,

genius_generous answered 1 year ago

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