1. Simple Machines

2. Simple Machines
Ordinary machines are typically complicated combinations of simple machines. There are six types of simple machines:
Simple Machine Example / description
Lever
Incline Plane
Wedge
Screw
Pulley
Wheel & Axle
crowbar
ramp
chisel, knife
drill bit, screw (combo of a wedge & incline plane)
wheel spins on its axle
door knob, tricycle wheel (wheel & axle spin together)

3. Simple Machines: Force & Work
By definition a machine is an apparatus that changes the magnitude or direction of a force. Machines often make jobs easier for us by reducing the amount of force we must apply, e.g., pulling a nail out of a board requires much less force is a pry bar is used rather than pulling by hand. However, simple machines do not normally reduce the amount of work we do! The force we apply might be smaller, but we must apply that force over a greater distance.
A complex machine, like a helicopter, does allow a human to travel to the top of a mountain and do less work he they would by climbing, but this is because helicopters have an energy source of their own (gasoline). Riding a bike up a hill might require less human energy than walking, but it actually takes more energy to get both a bike and a human up to the top that the human alone. The bike simply helps us waste less energy so that we don’t produce as much waste heat.

4. Force / Distance Tradeoff
Suppose a 300 lb crate of silly string has to be loaded onto a 1.3 m high silly string delivery truck. Too heavy to lift, a silly string truck loader uses a handy-dandy, frictionless, silly string loading ramp, which is at a 30º incline. With the ramp the worker only needs to apply a 150 lb force (since sin 30º = ½). A little trig gives us the length of the ramp: 2.6 m. With the ramp, the worker applies half the force over twice the distance. Without the ramp, he would apply twice the force over half the distance, in comparison to the ramp. In either case the work done is the same!
continued on next slide
150 lb
Silly String
300 lb
1.3 m
1.3 m
Silly String
30º

5. Force / Distance Tradeoff (cont.)
So why does the silly string truck loader bother with the ramp if he does as much work with it as without it? In fact, if the ramp were not frictionless, he would have done even more work with the ramp than without it.
answer: Even though the work is the same or more, he simply could not lift a 300 lb box straight up on his own. The simple machine allowed him to apply a lesser force over a greater distance. This is the “force / distance tradeoff.”
A simple machine allows a job to be done with a smaller force, but the distance over which the force is applied is greater. In a frictionless case, the product of force and distance (work) is the same with or without the machine.

6. Simple Machines & Potential Energy
Why can’t we invent a machine that decreases the actual amount of work needed to do a job?
answer: It all boils down to conservation of energy. In our silly string example the crate has the same amount of gravitational potential energy after being lifted straight up or with the ramp. The potential energy it has only depends on its mass and how high it’s lifted. No matter how we lift it, the minimum amount of work that a machine must do in lifting an object is equivalent to the potential energy it has at the top. Anything less would violate conservation of energy. In real life the actual work done is greater than this amount.
150 lb
Silly String
300 lb
1.3 m
1.3 m
Silly String
30º

7. Mechanical Advantage Fout M.A. = Fin
Mechanical advantage is the ratio of the amount of force that must be applied to do a job with a machine to the force that would be required without the machine. The force with the machine is the input force, Fin and the force required without the machine is the force that, in effect, we’re getting out of the machine, Fout which is often the weight of an object being lifted.
Fout
M.A. =
Fin
With the silly string ramp the worker only had to push with a 150 lb force, even though the crate weighed 300 lb. The force he put in was 150 lb. The force he would have had to apply without the ramp was 300 lb. Therefore, the mechanical advantage of this particular ramp is (300 lb) / (150 lb) = 2.
Note: a mechanical advantage has no units and is typically > 1.

8. Ideal vs. Actual Mechanical Advantage
When friction is present, as it always is to some extent, the actual mechanical advantage of a machine is diminished from the ideal, frictionless case.
Ideal mechanical advantage = I.M.A. = the mechanical advantage of a machine in the absence of friction.
Actual mechanical advantage = A.M.A. = the mechanical advantage of a machine in the presence of friction.
I.M.A. > A.M.A, but if friction is negligible we don’t distinguish between the two and just call it M.A.
I.M.A’s for various simple machines can be determined mathematically. A.M.A’s are often determined experimentally since friction can be hard to predict (such as friction in a pulley or lever).

9. I.M.A. vs. A.M.A. Sample
Let’s suppose that our silly string loading ramp really isn’t frictionless as advertised. Without friction the worker only had to push with a 150 lb force, but with friction a 175 lb force is needed.
Thus, the I.M.A. = (300 lb) / (150 lb) = 2, but the A.M.A. = (300 lb) / (175 lb) = 1.71.
Note that with friction the worker does more work with the ramp than he would without it, but at least he can get the job done.
175 lb
Silly String
1.3 m
300 lb
30º

10. I.M.A. for a Lever Fout dF I.M.A. = = Fin do dF d0 Fout Fin
A lever magnifies an input force (so long as dF > do). Here’s why: In equilibrium, the net torque on the lever is zero. So, the action-reaction pair to Fout (the force on the lever due to the rock) must balance the torque produced by the applied force, Fin. This means Fin· dF = Fout· do
Fout dF
Therefore,
I.M.A. = =
Fin do
do = distance from object to fulcrum dF = distance from applied force to fulcrum dF d0
Fout
Fin
fulcrum

11. I.M.A. for an Incline Plane
The portion of the weight pulling the box back down the ramp is the parallel component of the weight, mg sin. So to push the box up the ramp without acceleration, one must push with a force of mg sin. This is Fin. The ramp allows us to lift a weight of mg, which is Fout. So,
I.M.A. = Fout / Fin = mg / (mg sin ) = 1 / sin = d / h
This shows that the more gradual the incline, the greater the mechanical advantage. This is because when  is small, so is mg sin. d is very big, though, which means, with the ramp, we apply a small force over a large distance, rather than a large force over a small distance without it. In either case we do the same amount of work (ignoring friction). m d h 

12. M.A. for a Single Pulley #1 Fout m Fin
With a single pulley the ideal mechanical advantage is only one, which means it’s no easier in terms of force to lift a box with it than without it. The only purpose of this pulley is that it allows you to lift something up by applying a force down. It changes the direction, not the magnitude, of the input force.
Fout m
The actual mechanical advantage of this pulley would be less than one, depending on how much friction is present.
Fin mg
Pulley systems, with multiple pulleys, can have large mechanical advantages, depending on how they’re connected.

13. M.A. for a Single Pulley #2 Fin = F F F m
With a single pulley used in this way the I.M.A. is 2, meaning a 1000 lb object could be lifted with a 500 lb force. The reason for this is that there are two supporting ropes. Since the tension in the rope is the same throughout (ideally), the input force is the same as the tension. The tension force acts upward on the lower pulley in two places. Thus the input force is magnified by a factor of two. The tradeoff is that you must pull out twice as much rope as the increase in height, e.g., to lift the box 10 feet, you must pull 20 feet of rope. Note that with two times less force applied over twice the distance, the work done is the same.
Fin = F F F m mg

14. M.A: Pulley System #1 Fin = F F F F m
In this type of 2-pulley system the I.M.A. = 3, meaning a 300 lb object could be lifted with a 100 lb force if there is no friction. The reason for this is that there are three supporting ropes. Since the tension in the rope is the same throughout (ideally), the input force is the same as the tension. The tension force acts upward on the lower pulley in three places. Thus, the input force is magnified by a factor of three. The tradeoff is that you must pull out three times as much rope as the increase in height, e.g., to lift the box 4 feet, you must pull 12 feet of rope. Note that with three times less force applied over a three times greater distance, the work done is the same.
Fin = F F F F m mg

15. I.M.A: Pulley System #2 F F F Fin = F 1. Number of pulleys:
2. Number of supporting ropes:
3. I.M.A. =
4. Force required to lift box if no friction:
5. If 2 m of rope is pulled, box goes up:
6. Potential energy of box m up:
7 a. Work done by input force to lift box m up with no friction: 7 b. Work done lifting box m straight up without pulleys: If the input force needed with friction is 26 N,
9. A.M.A. =
10. Work done by input force now is:
3, but this doesn’t matter
3, and this does matter
3, since there are 3 supporting ropes
20 N
0.667 m F
40 J F F
20 N · 2 m = 40 J
60 N · m = 40 J
60 N
(60 N) / (26 N) = < I.M.A.
60 N
Fin = F
26 N · 2 m = 52 J

16. Efficiency = = Wout eff = = Win Note that in the last problem:
Work done using pulleys (no friction)
Work done lifting straight up
Potential energy at high point = =
little force × big distance
big force × little distance
m g h
All three of the above quantities came out to be 40 J. When we had to contend with friction, though, the rope still had to be pulled a “big distance,” but the “little force” was a little bigger. This meant the work done was greater: 52 J. The more efficient a machine is, the closer the actual work comes to the ideal case in lifting: m g h. Efficiency is defined as:
Wout
work done with no friction (often m g h)
eff = =
Win
work actually done by input force
In the last example eff = (40 J) / (52 J) = 0.769, or 76.9%. This means about 77% of the energy expended actually went into lifting the box. The other 13% was wasted as heat, thanks to friction.

17. Efficiency & Mechanical Advantage
Efficiency always comes out to be less than one. If eff > 1, then we would get more work out of the machine than we put into it, which would violate the conservation of energy. Another way to calculate efficiency is by the formula:
A.M.A.
To prove this, first remember that Wout (the work we get out of the machine) is the same as Fin × d when there is no friction, where d
eff =
I.M.A.
is the distance over which Fin is applied. Also, Win is the Fin × d when friction is present.
A.M.A.
Fout / Fin w/ friction
Fin w/ no friction = =
I.M.A.
Fout / Fin w/ no friction
Fin w/ friction
d Fin w/ no friction
Wout
In the last pulley problem, I.M.A. = 3, A.M.A. = = = =
eff
d Fin w/ friction
Win
Check the formula: eff = / 3 = 76.9%, which is the same answer we got by applying the definition of efficiency on the last slide.

18. Wheel & Axle Fin R = Fout r I.M.A. = Fout / Fin = R / r
Unlike the pulley, the axle and wheel move together here, as in a doorknob. (In a pulley the wheel spins about a stationary axle.) If a small input force is applied to the wheel, the torque it produces is Fin R. In order for the axle to be in equilibrium, the net torque on it must be zero, which means at the other end Fout will be large, since the radius there is smaller. Balancing torques, we get:
Fin R = Fout r
I.M.A. = Fout / Fin = R / r
With a wheel and axle a small force can produce great turning ability. (Imagine trying to turn a doorknob without the knob.) Note that this simple machine is almost exactly like the lever. Using a bigger wheel and smaller axle is just like moving the fulcrum of a lever closer to object being lifted. R r

19. Wheelbarrow as a Lever M.A. = dF / do
Schmedrick decides to take up sculpting. He hauls a giant lump of clay to his art studio in a wheelbarrow, which is a lever / wheel & axle combo. Unlike a see-saw, both forces are on the same side of the fulcrum. Since Fin is further from the fulcrum, it can be smaller and still match the torque of the load.
Acme Lump o’ Clay
Fin
Lever
fulcrum
Fout = mg
M.A. = dF / do d0 dF

20. Fbicep(4 cm) = (4 lb) (14 cm) + (40 lb) (30 cm)
Human Body as a Machine
The center of mass of the forearm w/ hand is shown. Their combined weight is 4 lb.
Fbicep
tendon
bicep
40 lb dumbbell
humerus
radius
4 lb
4 cm
ligament
c.m.
40 lb
14 cm
Because the biceps attach so close to the elbow, the force it exerts must be great in order to match the torques of the forearm’s weight and dumbbell:
30 cm
Fbicep(4 cm) = (4 lb) (14 cm) + (40 lb) (30 cm)
Fbicep= 314 lb !
continued on next slide

21. Human Body as a Machine (cont.)
Let’s calculate the mechanical advantage of this human lever:
Fout / Fin = (40 lb) / (314 lb) = 0.127
Fbicep
Note that since the force the biceps exert is less than the dumbbell’s weight, the mechanical advantage is less than one. This may seem pretty rotten. It wouldn’t be so poor if the biceps didn’t attach so close to the elbow. If our biceps attached at the wrist, we would be super duper strong, but we wouldn’t be very agile!
4 lb
4 cm
40 lb
14 cm
30 cm