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Brandon T.
Mechanical Engineering Student with Research Background
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Material Science
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Question:

Tensile stress is applied to a single crystal of nickel along a  direction. A slip occurs on a (1 1 1) plane and in a [-1 0 1] direction at a minimum applied stress of 13.9 MPa or 2020 psi. Find the critical resolve shear stress

Brandon T.

To start, draw a 3D axis and the vectors for the direction of the force [F], the direction of the slip [S] and the direction of the plane (the normal vector of the plane 'n'). The critical resolve shear stress is defined by ϕ (the angle between the force and the plane vectors) and λ (angle between the force and the slip vectors) and is equal to the stress where the material yielded times cosϕ*cosλ. The yield stress is given as 13.9 MPa so all that's left is to find those two angles. The values for cosϕ and cosλ can be found by rewriting the dot product of 2 vectors A•B=|A|*|B|*cos(angle between them). As a result you get F•S/( ||F|| * ||S|| )=cosϕ and F•n/( ||F|| * ||n|| )=cosλ. Knowing that a dot product is the sum of the products of the i j k vector components of each vector, the solution can be found. The final answer the the problem is 5.68MPa

Calculus
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Question:

A triangular solid of revolution is developed by rotating a triangular plate through 2 pi radians. The result is a "washer" with an inner radius of 'a' an outer radius of 'b' and a height of 2 times 'c'. Derive a general equation for the volume of the solid.

Brandon T.

Best way to approach a problem like this is to draw a 2-D cross sectional diagram and label the dimensions with the variables 'a' 'b' and 'c' as stated in the problem. With this you can create an 'x' and a 'y' axis at the center to form a graph and solve the problem using the "washer" method of revolution (2𝜋 ∫r*h*dr). To make things simpler you can consider only revolving the top-right quadrant of the graph so that you can create a line function where the slope is c/(b-a) and the y intercept is the slope times -a. Thus the function for the line is y = c/(b-a)x - c/(b-a)*a or c/(b-a)*(x-a). Using this line we can now use the washer method and integrate using 2𝜋 ∫x*y(x)*dx (where x = r, y(x) = h and dx = dr) and the limits of integration are from 'a' to 'b'. The resulting volume should come out to 1/3𝜋*c*(2b^2 - ab - a^2). This value is the top half of the shape since it was found using the top half of the original triangle. After multiplying by 2, the final answer is 2/3𝜋*c*(2b^2 - ab - a^2).

Physics
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Question:

A man in a rocket flies away from earth at a speed of 93,000 miles per second on January 23rd, in the year 2010. After miraculously surviving the tremendous force exerted on his body from the launch, he stays in the rocket until his clock reads that exactly 5 years has passed. After returning to Earth he asks a stranger for the date. Assuming this stranger doesn't lie, what is the date he is told?

Brandon T.

Due to relativity and since 2012 was a leap year, the stranger would tell the man that it's time to go costume shopping because it's Halloween in the year 2015. Otherwise known as October 31st, 2015.

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