1.
Review
the results of Exam # 2. Focus on
the multiple-choice questions and on Problem 11.

2.
The Egyptian pyramids,
made of a total of about 25 million tons of limestone, were built entirely with
human labor. To get some insight
into how this might have been accomplished, consider lifting a
"small" pyramid stone with a weight of 20,000 N, which is about 2
tons (some of the larger stones actually weigh up to 70 tons).

a) **Estimate**
the number of able-bodied people it would take to lift such a stone off the
ground. Begin by deciding how much
a typical person (such as you) can lift.

Now consider lifting the stone with the following
arrangement, consisting of two long levers resting on sturdy pivot points
(sometimes called fulcrums). The
downward arrows represent the forces applied by the workers.

b) Make a free body diagram of one of the levers as it is
lifting the stone. You can neglect
the weight of the lever itself.

c) Write out the torque equation for the lever, choosing
the fulcrum as the axis of rotation.

d) How much force must be applied to the end of each
lever in order to lift the stone at a steady rate?

Other things to consider:

*Look carefully at the sketch of
the levers and the block. YouÕll
see that the distance between each fulcrum and the end of each lever is not
exactly the same as the distance between the fulcrum and the edge of the
block. Which distance should be
used as the lever arm of the force exerted by the block on the lever, and why?*

*What might the limitations of a
simple lever system be?*

*Can you determine the force
exerted on the lever by the fulcrum?*

* *

* *

3. Consider a ladder of mass M and length L leaning
against a wall at some angle q, as shown in the sketch to the right. The bottom of the ladder is rubberized
(so there's a lot of friction), but the top is bare aluminum and can be
considered frictionless.

a) Make a free-body diagram for this ladder, including a
coordinate system.

b) Write out the equations for static equilibrium of the
ladder. (Indicate with a ÒPÓ on
your FBD the point you are choosing to sum torques around, and show with an
arrow your choice for the direction of a positive torque.)

c) In terms of the known quantities (M, L and q), determine all of the forces exerted on the ladder
by the wall and by the floor.

Other things to consider:

*Suppose someone is standing on the ladder. Which forces on the ladder do you
expect to be different, and will they be bigger or smaller? Show explicitly how your equations will
be modified.*

4.
For the same ladder
above suppose that the coefficient of static friction between the floor and the
ladder is known to be m = 1/2.

a) What is the maximum amount of friction that the floor
can exert on the ladder? Express
your result as a multiple of the weight of the ladder (examples: f = 2mg or f =
0.25mg).

b) At what angle q is the frictional force at its maximum?

c) What would happen to the ladder if q were made smaller?

Other things to consider:

*The coefficient of friction between the ladder and the
floor is a constant number. How
then do you explain (in words) the fact that the ladder becomes unstable as the
angle **q** is decreased?*

*Which is more likely to slip when leaned against the wall
as in this problem, a tall ladder or a short ladder? Assume that the ladders have the same rubber on the bottom.*