6-30. A column is made of two 1-in. by 9-in. and two 2-in. by 8-in. pieces of lumber arranged as shown in Fig. P6-30. Determine the radii of gyration with respect to the horizontal and vertical centroidal axes 12" of the cross section. 9" I2" 8" P6-30
Angular speed, acceleration and displacement
Angular acceleration is defined as the rate of change in angular velocity with respect to time. It has both magnitude and direction. So, it is a vector quantity.
Angular Position
Before diving into angular position, one should understand the basics of position and its importance along with usage in day-to-day life. When one talks of position, it’s always relative with respect to some other object. For example, position of earth with respect to sun, position of school with respect to house, etc. Angular position is the rotational analogue of linear position.
Given data
The length of the lumber is l = 8.”
The width of the lumber is y = 2.”
The length of the column is B = 9.”
The width of the column is a = 1.”
The labeled figure for the above cross section is shown below:
The lumber and column make the hollow cross section. The width and length of the inner cross section are given as:
The width and length of the outer structure are given as:
The expression for the area of the enclosed section is given as:
Substitute the value in the above equation.
The expression for the moment of inertia about the centroid axis of the cross section in the horizontal axis is given as:
Substitute the value in the above equation.
The expression for the moment of inertia about the centroid axis of the cross section in the vertical axis is given as:
Substitute the value in the above equation.
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