October 2012

Variation / Proportional

Direct Variation / Directly Proportional

y is directly proportional to x, y ∝ x:
 
$y = kx$

k = constant of proportionality
y varies directly as x is another statement equivalent to the above statement.

 

Inverse Variation / Directly Proportional

y is inversely proportional to x, y ∝ 1/x:
 
$y = \dfrac{k}{x}$

k = constant of proportionality

Three-digit numbers not divisible by 3

Problem
How many three-digit numbers are not divisible by 3?
 

720 Two triangles | Centroid of Composite Area

Problem 720
The centroid of the sahded area in Fig. P-720 is required to lie on the y-axis. Determine the distance b that will fulfill this requirement.
 

Centroid involving a triangle of unknown base

 

721 Increasing the width of flange to lower the centroid of inverted T-beam

Problem 721
Refer again to Fig. P-714. To what value should the 6-in. width of
the flange be changed so that the centroid of the area is 2.5 in. above the base?
 

Inverted T-section for centroid problem

 

722 Semicircle and quarter circle | Centroid of composite area

Problem 722
Locate the centroid of the shaded area in Fig. P-722 created by cutting a semicircle of diameter r from a quarter circle of radius r.
 

723 Rectangle, quarter circle and triangle | Centroid of Composite Area

Problem 723
Locate the centroid of the shaded area in Fig. P-723.
 

724 Rectangle, semicircle, quarter-circle, and triangle | Centroid of Composite Area

Problem 724
Find the coordinates of the centroid of the shaded area shown in Fig. P-724.
 

725 Centroid of windlift of airplane wing | Centroid of area

Problem 725
Repeat Problem 239 without using integration.
 

726 Area enclosed by parabola and straigh line | Centroid of Composite Area

Problem 726
Locate the centroid of the shaded area enclosed by the curve y2 = ax and the straight line shown in Fig. P-726. Hint: Observe that the curve y2 = ax relative to the y-axis is of the form y = kx2 with respect to the x-axis.
 

Moment of Inertia and Radius of Gyration

Moment of Inertia
Moment of inertia, also called the second moment of area, is the product of area and the square of its moment arm about a reference axis.
 

Moment of inertia about the x-axis:

$\displaystyle I_x = \int y^2 \, dA$

 

000_moment_of_inertia.gif

 

816 Polar moment of inertia and radius of gyration at one corner of rectangle

Problem 816
A rectangle is 3 in. by 6 in. Determine the polar moment of inertia and the radius of gyration with respect to a polar axis through one corner.
 

817 Hollow Tube | Moment of Inertia and Radius of Gyration

Problem 817
Determine the moment of inertia and radius of gyration with respect to a polar centroidal axis of the cross section of a hollow tube whose outside diameter is 6 in. and inside diameter is 4 in.
 

818 Hollow square section | Moment of Inertia and Radius of Gyration

Problem 818
A hollow square cross section consists of an 8 in. by 8 in. square from which is subtracted a concentrically placed square 4 in. by 4 in. Find the polar moment of inertia and the polar radius of gyration with respect to a z axis passing through one of the outside corners.