Problem 821
Find the moment of inertia about the indicated x-axis for the shaded area shown in Fig. P-821.
 

Problem 1
Show that the largest triangle of given perimeter is equilateral.
 

Let F be a function of several variables, say x, y, and z. In symbols,

The partial derivative of F with respect to x is denoted by

and can be found by differentiating f(x, y, z) in terms of x and treating the variables y and z as constants.
 

Problem 6
Eliminate the c₁ and c₂ from x = c₁ cos ωt + c₂ sin ωt. ω being a parameter not to be eliminated.
 

Problem 5
Eliminate A and B from x = A sin (ωt + B). ω being a parameter not to be eliminated.
 

Problem 856
For the beam shown in Fig. P-856, determine the moments over the supports. Also draw the shear diagram and compute the maximum positive bending moment.
 

Problem 855
If the distributed load in Prob. 854 is replaced by a concentrated load P at midspan, determine the moments over the supports.
 

Answers:
$M_1 = \dfrac{PL}{8} \cdot \dfrac{1}{\alpha + 2} = M_4$

$M_2 = -\dfrac{PL}{8} \cdot \dfrac{2}{\alpha + 2} = M_3$
 

Problem 854
Solve for the moment over the supports in the beam loaded as shown in Fig. P-854.
 

Answers:
$M_1 = \dfrac{w_o L^2}{12} \cdot \dfrac{1}{\alpha + 2} = M_4$

$M_2 = -\dfrac{w_o L^2}{12} \cdot \dfrac{2}{\alpha + 2} = M_3$
 

Problem 853
For the continuous beam shown in Fig. P-853, determine the moment over the supports. Also draw the shear diagram and compute the maximum positive bending moment. (Hint: Take advantage of symmetry.)