7) Calculate the maximum radius of a sphere that may be accommodated in an octahedral hole in a close packed solid composed of spheres of radius r. 8) Given a close-packed arrangement of identical spheres, calculate the percentage of the unoccupied space.
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- 4. Four identical bricks are stacked up on top of each other. Each has a length of a and a mass of m. What can be the maximum value of x? smooth surface a4. Certain frictional forces have magnitude proportional to the square of the velocity of the moving object. Thus |Ffriction| = -yv² for some positive constant y. Here there is an annoying sign issue since the frictional force must have sign opposite to the velocity. One can write Felastic = -y|v|v. If an object of mass m is subject to this frictional force and no other forces, and the initial velocity is positive, then the velocity will always be positive and we can just say Ffriction Then the position y of the particle satisfies the equation -yo?. dy dt2 2 dy +y dt = 0. Write out a solution of this equation for the initial conditions y(0) = Yo, (dy/dt)(0) the substitution u = = vo. Hint: The equation reduces to first order. Use dy/dt and solve the first order equation to get u. Then integrate u to get y.2.
- 1-A circular tube of outer diameter d;-90 mm and inner diameter d;=70 mm is welded at the right-hand end to a fixed plate and the left-hand end to a rigid end plate (see figure). A solid circular bar of diameter d;=60 mm is inside of and concentric with the tube. The bar passes through a hole in the fixed plate and is welded to plate the rigid endplate. The bar is 1.0 m long, and the tube is half as long as the bar. A torque T=1000 N. m acts at end A of the bar. The bar and tube are also made of an aluminum alloy with a shear modulus of elasticity G=30 GPa. (a) Determine the maximum shear stresses in both the bar and Tube Fired plate End Har Tube Har tube. (b) Determine the angle of twist (in degrees) at end A of the bar.Find the center of mass of a laminate bounded by y=−x,y=x,y=2 with variable density modeled by \rho (x,y)=y+5. Question 3 Answer a. (0,2619) b. (35,94) c. (−23,116) d. None of the above e. (0,1) f. (0,32)If the one of axies value was negative then move the center of cubic to a) Direct value direction b) Negative value direction C) origin value direction d) Possitive value direction
- 10. Calculate the second moment of area for a 2 x 6 piece of lumber about the x-x and y-y axes, as shown. y X y1. The setup shown by the diagram below was used to calculate the force experienced by a small mass m situated at a distance r from the geometric center of a spherical shell with mass M, radius R, and a wall thickness of t. M C dᎾ R 0 Rde A Rsine B r Ф dF Ф m If the mass is situated outside the spherical shell, the force can be calculated by integration from l = (r - R) to l= (r+ R), as shown here: l=r+R R - [ &F="TOmp: +(1+²²=² ) We SdF=fGmpat- de l=r-R What would the limits of integration be if the mass were situated inside the spherical shell? Include a labeled diagram to represent this situation.2. You are manufacturing a can which is a perfect cylinder. The ideal volume of the can is 375 cm3. Since the bases of the cylinders are exactly 50 cm2, the volume of the can is V = 50h where h is the height of the can in centimeters. You need to cut the sides of the can so the volume has at most an error of 10 cm³. Give the range of heights the can could have and still meet the required specifications for its volume.