Work of Forces Acting On A Rigid Body
The work of forces acting at various points on a single rigid body can be calculated from the work of a resultant force and torque. To see this, let the forces F1, F2 ... Fn act on the points X1, X2 ... Xn in a rigid body.
The trajectories of Xi, i=1,...,n are defined by the movement of the rigid body. This movement is given by the set of rotations and the trajectory d(t) of a reference point in the body. Let the coordinates xi i=1,...,n define these points in the moving rigid body's reference frame M, so that the trajectories traced in the fixed frame F are given by
The velocity of the points Xi along their trajectories are
where ω is the angular velocity vector obtained from the skew symmetric matrix
known as the angular velocity matrix.
The small amount of work by the forces over the small displacements δri can be determined by approximating the displacement by δr=vδt so
or
This formula can be rewritten to obtain
where F and T are the resultant force and torque applied at the reference point d of the moving frame M in the rigid body.
Read more about this topic: Work (physics)
Famous quotes containing the words work, forces, acting, rigid and/or body:
“I work all day, and get half-drunk at night.
Waking at four to soundless dark, I stare.
In time the curtain-edges will grow light.
Till then I see what’s really always there:
Unresting death, a whole day nearer now,
Making all thought impossible but how
And where and when I shall myself die.”
—Philip Larkin (1922–1986)
“The next thing his Lordship does, after clearing of the coast, is the dividing of his forces, as he calls them, into two squadrons, one of places of Scriptures, the other of reasons....
All that I have to say touching this, is that I observe a great part of those his forces do look and march another way, and some of them fight amongst themselves.”
—Thomas Hobbes (1579–1688)
“It especially helps if you know that we’re all faking our adulthood—even your parents and their parents. Beneath these adult trappings—in our president, in our parents, in you and me—lurk the emotions of a child. If we know that only about ourselves, we become infantile; if we understand that about everybody, then we have nothing to be ashamed of—unless, of course, we go around acting like a child and expecting everyone else to act like grownups.”
—Frank Pittman (20th century)
“I found a dimpled spider, fat and white,
On a white heal-all, holding up a moth
Like a white piece of rigid satin cloth—
Assorted characters of death and blight”
—Robert Frost (1874–1963)
“...we would rather be away from the body and at home with the Lord.”
—Bible: New Testament, 2 Corinthians 5:8.