Lever and Leverage By： Yi Zheng Chengwei Ge Jinxin Chen Yaqi Yang Qian Xiao

In the lecture 07B we learned Torque In the lecture 07B we learned Torque. Once we saw Torque, lever appeared in our mind, because there is a principle called: The 2rd Principle of Leverage. This really interests us. Hence, this time we are going to tell something about Lever, Leverage and these principles. Before the introduction of how lever and torque related to each other, let’s find out what is lever first…

What is lever In physics A lever is a rigid object that is used with an appropriate fulcrum or pivot point to multiply the mechanical force /effort that can be applied to another object/load. Information from Wikipedia http://en.wikipedia.org/wiki/Lever

Category of the lever Then, we learned from website that lever has three classes. These classes represent variation in the relative locations of the fulcrum, the load and the force. Information from Wikipedia http://en.wikipedia.org/wiki/Lever

The first Class: The fulcrum is located between the applied force and the load. In this class, seesaw, crowbar and scissors are all great examples. Information from Wikipedia http://en.wikipedia.org/wiki/Lever

The second Class: The load is situated between the fulcrum and the force. Wheelbarrow and nutcracker can clearly explain this class’s characteristic. Information from Wikipedia http://en.wikipedia.org/wiki/Lever

The third Class: The force is applied between the fulcrum and the load. Tweezers and mandible are under this class. Information from Wikipedia http://en.wikipedia.org/wiki/Lever

However those three classes seemed not that pragmatic in our daily life. So, after further study, we found another classification of lever, this method is obviously easier to understand… According to the different effects lever has, the lever can also be divided into another three types. Respectively, they are: saving-power lever; laborious-power lever; equal-power lever

Saving-power lever Definition: the lever which has the power greater than the resistance arm of the lever arm Advantage: save force and power Disadvantage: need relative long distance

Equal-power lever Definition: the lever which has the force equal to the resistance arm of the lever arm Advantage: ------ Disadvantage:------ Notice: Since this kind of lever is neither save the power nor save the distance, we are not sure what are its strong points and drawbacks.

Laborious-power lever Definition: the lever which has power less than the resistance arm of the lever arm Advantage: short distance Disadvantage:   laborious

Then, It comes to the LEVERAGE…

Leverage Leverage is also been called as “the equilibrium conditions of lever”. To make lever attain equilibrium state, the number of forces acted upon on the two points of the lever must have inverse relationship with their force arms

Mathematical Expression Mechanical force × Power arm = Resistant force × Resistance arm Using algebra to express it as F1 • L1 = F2 • L2 In this expression, F1 stands for mechanical force L1 stands for power arm, F2 represents resistant force L2 represents resistance arm

After understanding what is lever and leverage After understanding what is lever and leverage. It is high time to turn the page and learn those marvelous principles created by physicists.

Introductory to the principle & History background Archimedes, ancient Greek scientist, once said “give me a fulcrum and I will be able to lever up the whole earth” The portrait of Archimedes

Archimedes’ Achievement : Archimedes in On the plane figure of the balance was first presented in his book leverage.  He began to leverage some experience in the practical application of knowledge as a "self-evident axiom", and then proceed from these axioms, the use of geometry through the strict logical demonstration, come to the lever principle

These axioms are: (1) without the weight of the rod ends of equal distance from the pivot mount equal weight, they will balance (2) without the weight of the rod ends at equal distances from the pivot mount not equal the weight of the heavy end of the declination

(3) without the weight of the rod ends are not equal distance from the fulcrum of an equal weight hanging from the far end of the declination (4) the role of a weight can be uniformly distributed with a few to replace the role of weight, as long as the center of gravity remains unchanged. Instead of several uniform distribution of weight can be a hanging in their center of gravity at the place of weight

(5) similar to the graph of the distribution center is in a similar manner starting from these axioms In addition, there is a conclusion that in the "center of gravity" theory, based on the Archimedes discovered the principle of lever, that "two weights are balanced, they are away from the fulcrum is inversely proportional to the distance and weight.”

Lever of Archimedes not only stayed on the theoretical aspects, but also carried out a principle whereby series of inventions. It is said that he had the help of levers and pulleys, so parked in the beach into the water on masts as well, in the defense of Syracuse against the Roman naval attack in the battle, the principle of Archimedes created a much leverage, close of the sling, use it to shoot a variety of missiles and stones to attack the enemy, once outside the city the Romans arrested at Syracuse for 3 years.

Moreover… we find the relationship between torque and lever: Due to the foundation of principle of leverage, there develops another principle: the second principle of lever.

Explanation of The Second Principle of Lever the second principle of lever which is used in the 3D space. There exists two torques in the bowed lever in the process of rotating. One is power torque, the other is magnified inner resistance torque. The theory of first principle of lever all comes from the statics while the second principle of lever is applied in the field of dynamics.

Using algebra to express it as : M1×r2×L1 = M2 ×r1×L2 In which: M1 is moment torque, M2 is resistance torque, L1 is power arm, L2 is resistance torque, r1 is radius of moment torque, r2 is radius of the inner resistance torque

The second principle of leverage is created to do the problems that the first principal cannot solve. For example: we can use the 2rd principle of leverage to solve the hard nut called the real long-term CVT when dealing with the mechanical transmission.

This principle establishes a contact between lever and torque This principle establishes a contact between lever and torque. Although it is very useful, the application of this principle is far beyond our knowledge … Compared with this one, the following application of lever is much easier …

Application 2: Lever in our body Muscles and bones act together to form levers… as a matter of fact, laborious-power levers occupy human body’s levers in a great measure, and the others are equal-power and saving-power lever.

Class 1 lever: nod head When you nod your head, the pivot is the place where the skull meets the top of the spine. The lever arm is your skull. You use your the neck muscles to surmount the load caused by the weight of the head and lift your head up.

Class 2 lever: Stand on the tip toes When you stand on your tip toes, the tip toe is the pivot and the muscle on your heel exerts a pulling force upward, equaling the effect of your weight’s gravity of your weight. Since the pulling force of muscle is less than the weight, this set is a laborious-power lever. Besides, the longer your feet are, the less force you need to spend.

Class 3 lever: Bent Arm During the process of lifting up heavy goods, arm acts as the lever and elbow point is the pivot. This set is a laborious-power lever. Usually, people need to spend six more strength to lift up the goods; however, it can save the distance.

So far, we have done lots of research and found out the amazing relationship between torque and lever. We also use this opportunity to learn the lever and leverage thoroughly. That’s the end of this project… THANKS!!!

Reference : Wikipedia http://en.wikipedia.org/wiki/Lever Baidu Baike http://baike.baidu.com/view/85069.htm Google image http://www.google.com/images?q=%E6%9D%A0%E6%9D%86%E5%8E%9F%E7%90%86&hl=en&tbas=0&prmd=iv&source=lnms&tbs=isch:1&ei=1RL3TPX_Csmr8AaF4Mm8BQ&sa=X&oi=mode_link&ct=mode&ved=0CA4Q_AU&biw=1639&bih=810