Find the radius of gyration of a plate covering the region bounded by x=3, x=5, y=0, y=2 with respect to the y-axis.

 

Answer:

Radius of gyration = 4.39

Step-by-step explanation

Horzontal length of plate = 5 - 3 = 2

Vertical length = 2 - 0 = 2

Area of plate = Horizontal length X Vertical length

= 2 X 2

= 4

The plate can be thought of as made of many thin strips parallel to its vertical length.

Let there be a thin vertical strip at distance x from y-axis. The strip is parallel to y-axis, with horizontal thickness dx and vertical length = 2.

Area of this strip = 2dx

Distance of all points on strip from y-axis = x

Hence, moment of inertia of strip with respect to y-axis

d?I? = Area X (distance)²

d?I? = 2x² dx

Integrating, we get moment of inertia about y-axis.

The limits will be x = 3 to x = 5.

?       5

I? = ?∫3?? 2x² dx

= 2x³ / 3

= [ 2(5)³ - 2(3)² ] / 3

= [ 250 - 18 ] / 3

= 77.33

Hence, radius of gyration = ( Moment of inertia / Area of plate )^1/2

= (77.33 / 4)^1/2

= ?under root19.33??

= 4.39