Simple Nature - Light and Matter

t/Example 68.

u/Archimedes’ principle works

regardless of whether the object

is a cube. The fluid makes a force

on every square millimeter of the

object’s surface.

three vectors. This is the Pythagorean theorem, which will hold
only if the three vectors form a right triangle.
The fact that we observe the angle to be somewhat less than
90 degrees shows that the assumption used in the proof is only
approximately valid: a little energyisconverted into heat and
sound. The opposite case would be a collision between two blobs
of putty, where the maximum possible amount of energy is con-
verted into heat and sound, the two blobs fly off together, giving
an angle of zero between their momentum vectors. The real-life
experiment interpolates between the ideal extremes of 0 and 90
degrees, but comes much closer to 90.
Force is a vector, and we add force vectors when more than one
force acts on the same object.

Pushing a block up a ramp example 68

.Figure t/1 shows a block being pushed up a frictionless ramp

at constant speed by an applied forceFa. How much force is

required, in terms of the block’s mass,m, and the angle of the

ramp,θ?

.We analyzed this simple machine in example 38 on page 172

using the concept of work. Here we’ll do it using vector addition

of forces. Figure t/2 shows the other two forces acting on the

block: a normal force,Fn, created by the ramp, and the gravita-

tional force,Fg. Because the block is being pushed up at constant

speed, it has zero acceleration, and the total force on it must be

zero. In figure t/3, we position all the force vectors tip-to-tail for

addition. Since they have to add up to zero, they must join up

without leaving a gap, so they form a triangle. Using trigonometry

we find

Fa=Fgsinθ

=mgsinθ.

Buoyancy, again example 69

In example 10 on page 85, we found that the energy required to

raise a cube immersed in a fluid is as if the cube’s mass had been

reduced by an amount equal to the mass of the fluid that other-

wise would have been in the volume it occupies (Archimedes’

principle). From the energy perspective, this effect occurs be-

cause raising the cube allows a certain amount of fluid to move

downward, and the decreased gravitational energy of the fluid

tends to offset the increased gravitational energy of the cube. The

proof given there, however, could not easily be extended to other

shapes.

Thinking in terms of force rather than energy, it becomes easier

to give a proof that works for any shape. A certain upward force is

Section 3.4 Motion in three dimensions 207