A weight lifter picks up a barbell and

1. lifts it chest high
2.holds it for 30 seconds
3.puts it down slowly (but does not drop it)

Rank the amounts of work W the weight lifter performs during these three operations. Label the quantities as W1, W2, and W3. Justify your ranking order. (Hint: Think about how work is defined in terms of who is applying forces and who is doing work.)

From highest to lowest:

Explanation:

The work done by a force is given by

where:

F is the force applied on the object

d is the displacement of the object

is the angle between the direction of the force and of the displacement

In case 1), the barbell is lifted upward. This means that:

- the force applied is upward

- the displacement of the object is upward

This means that , so the work done is positive:

In case 2), the barbell is held stationary: this means that the displacement is zero,

And therefore, this means that the work done is also zero:

In case 3), the barbell is put down slowly, without dropping it. This means that:

- The force applied is still upward (in fact, the force applied must be upward in order to overcome the force of gravity downward, and avoid the barbell to fall down)

- The displacement of the barbell is downward

This means that , so , and therefore the work done is negative:

So the ranking from greatest to smallest work is

i think it because equal matrices have equal dimensions, only square matrices can be symmetric. for all indices and . every square diagonal matrix is symmetric, since all off-diagonal elements are zero. in linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space.