Download I need to show mathematically how two 3rd-class levers

I need to show mathematically how two 3rd-class levers joined together in a series increase angular velocity versus
using a single 3rd-class lever. For example, a football player kicking the ball from the tee. I need to compare when he
kicks it with his kicking leg maintained in a straight condition, i.e. only his hip joint rotating (keeping his knee locked
allowing no rotation at the knee)..as a single 3rd-class lever, versus, more typically, kicking the ball normally by
allowing both his hip and knee to rotate during the kicking event. In other words, I need to show how the angular
velocity of the kicker's foot is increased when he rotates both knee and hip during the kicking action and how the two
levers provide more mechanical advantage over a single lever (hip rotation only scenario).
In a third class lever the load is located at the end of the lever. The effort is
exerted between the load and the fulcrum. The effort expended is greater than
the load, but the load is moved a greater distance. In other words, effort is
sacrificed in order to gain distance.
Load X Load Arm= Effort X Effort Arm
Load arm= distance between load and fulcrum
Effort arm= distance between effort and fulcrum
In a third class lever the effort arm is always less than the load arm (effort is
between the load and the fulcrum). Hence the effort is always greater than the
load. But the load moves a greater distance than the effort.
TO PROVE
Two 3rd-class levers joined together in a series increase angular velocity versus using a
single 3rd-class lever.
CASE I
Single third class lever:
Football player kicking the ball from the tee when he kicks it with his kicking leg
maintained in a straight condition:
The fulcrum is the hip joint.
The effort is applied by the thigh muscles (muscles moving the femur).
Effort arm= distance between the hip and the thigh muscles. =R hip-thigh
Load arm= distance between the foot and the hip joint. = R hip-foot
Effort= Effort of the thigh muscles= E thigh
Load= Kick
Load X Load Arm= Effort X Effort Arm
Or Kick X R hip- foot- = E thigh X R hip-thigh -----------------------(1)
Torque which supplies the angular momentum and hence angular velocity is equal to
force X distance= Kick X R hip- foot which in turn equal to E thigh X R hip-thigh .
Thus torque= E thigh X R hip-thigh -------(A)
CASE II
two 3rd-class levers joined together in a series:
Kicking the ball normally by allowing both his hip and knee to rotate during the kicking
event
Now there are two third class levers:
First lever
Fulcrum is again as before the hip joint.
Effort is applied by the thigh muscles(muscles moving the femur) as before.
However load is at the knee.
Effort arm= distance between the hip and the thigh muscles. =R hip-thigh
Load arm= distance between the hip joint and the knee = R hip-knee
Effort= Effort of the thigh muscles= E thigh
Load= Force applied at knee
Load X Load Arm= Effort X Effort Arm
Or Force applied at the knee X R hip-knee = E thigh X R hip-thigh ----------------------(2)
Torque which supplies the angular momentum and hence angular velocity is equal to
force X distance= Force applied at the knee X R hip-knee which in turn equal to E thigh X
R hip-thigh.
Thus torque= E thigh X R hip-thigh as in the case of one third class lever alone (equation A
above.) ------------------(B)
Second lever
Fulcrum is the knee joint.
Effort is applied by the muscles in the shin ( muscles moving the tibia and fibula) .
Load is at the foot.(foot kicks the ball)
Effort arm= distance between the knee and the shin muscles. =R knee-shin
Load arm= distance between the knee joint and the foot = R knee-foot
Effort= Effort of the shin muscles= E shin
Load= Kick
Load X Load Arm= Effort X Effort Arm
Or Kick X R knee-foot = E shin X R knee-shin ----------------------(3)
Torque which supplies the angular momentum and hence angular velocity is equal to
force X distance= Kick X R knee-foot which in turn equal to E shin X R knee-shin.
Thus torque= E shin X R knee-shin -----------------(C)
Thus in the case of two third class levers the torque=B+C
CONCLUSION
As we have shown B= A (Assuming effort applied by the thigh muscles is the same)
Thus in the case of two levers working together (CASE II) the torque is greater than
when only one lever is acting (CASE I) by the amount C.
Angular velocity which depends on torque is therefore greater in the CASE II.
Thus it is proved that two 3rd-class levers joined together in a series increase
angular velocity versus using a single 3rd-class lever.