Slajd line of force

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					TORQUE, CENTER OF MASS, CENTER OF GRAVITY
                Semester I
                2010/2011
    Motion in which an entire object moves is called translation




 Motion in which an object spins is called rotation




The point or line about which an object turns is its center of rotation
The LEVER ARM, d, is the
perpendicular distance from the
axis of rotation to the line of
action of the force.
   d = L sin Φ




   FULCRUM - the point of support,
   or axis, about which a lever may be made to rotate
      How does force create rotation ??



A torque is an action that causes objects
to rotate.
Torque is not the same thing as force.
         Torque
  rF
                                                 F


                                         r
TORQUE is the cross product between the distance vector
(the distance from the pivot point to the point where force is applied)
and the force vector




 magnitude :   rF sin 
               rF sin 
To make an object rotate, a force must be applied in
the right place.

                                       0



                                        0
         Torque

         rF

magnitude :   rF sin 

          Unit Nm
 Consider force required to open door.

Is it easier to open the door by pushing/pulling
  away from hinge or close to hinge?

CLOSE?                                   AWAY?
The amount of torque depends on where and in what direction the
force is applied, as well as the location of the axis of rotation.




                                              What property
                                              of the applied
                                             force causes the
                                              door to open?




                rF sin 
Using the concept of torque, explain why the “easy-off” cap is easier
to unscrew than the normal cap.
          Some Physics Quantities
Vector - quantity with both magnitude (size) and direction
Scalar - quantity with magnitude only


    Vectors:                     Scalars:
    •Velocity                    •Energy
    • Force                      •Work
               Cross product
 The cross product of two vectors a and b is defined as

c  ab                         rF
 whose magnitude is

c  ab sin                      rF sin 
where  is the angle (< 180o) between a and b,


 whose direction is perpendicular to both a and b
 in the sense of the right-hand rule.
   Cross product (right-hand rule )


c  ab         Use the right-hand rule to
                determine the direction of the
                resulting vector in a cross
                product. Hold your right hand in
                front of you so that the thumb is
                pointed up, the index finger is
                pointed away from you and the
                middle finger is pointed to your
                left. The index finger shows the
                direction of vector A, the middle
                finger shows the direction of
                vector B and the thumb shows
                the direction of the vector from
                the cross product A x B.
 •   A torque ( a vector quantity) that tends o produce
     a counterclockwise rotation is considered positive.

 •   A torque that tends to produce a clockwise rotation
     is negative.



By convention, the sign of torque is:
<0       clockwise

>0                counter-clockwise
 A force of 50 newtons is applied to a wrench that is 30
 centimeters long. Calculate the torque if the force is applied
 perpendicular to the wrench so the lever arm is 30 cm




  rF sin   (0.3m)(50 N ) sin 90                o


  rF sin   (0.3m)(50m)(1)  15Nm
For the same force, you get more torque with a bigger
wrench  the job is easier!

 The torque can be increased by applying the force at right angles to
 the lever arm or by extending the lever arm.
For your arm, leg or any body part to move the appropriate
muscles and bones must work together as a series of levers.
A lever comprises of three components :

Fulcrum or pivot - the point about which the lever rotates

Load - the force applied by the lever system

Effort - the force applied by the user of the lever system

The way in which a lever will operate is dependent on the type of
lever.
Levers are important in human
motion because the human
body is a system of levers.

Our joints are axes of rotation
(fulcrums) and our bones are
the levers. Forces to move the
levers are provided by our
muscles, gravity, and external
forces.
       Three Classes of Levers

First Class - fulcrum
between Input and output


Second Class – output
between fulcrum and input


Third Class – input
between fulcrum and
output
First Class                 Second Class




              Third Class
Mechanical Equilibrium
  • To ensure that an object
    does not accelerate or
    rotate two conditions
    must be met:
  • net force = 0
  • net torque = 0
    Mechanical Equilibrium
•    First Condition of Equilibrium
            • The net external force must be zero

                          F  0
                Fx  0 and Fy  0
      – This is a statement of translational equilibrium

•    Second Condition of Equilibrium
          • The net external torque must be zero


                          0
           • This is a statement of rotational equilibrium
            SEESAW

       2m            2m




200N


                          400N
Unbalanced torques
                    balanced torques

                   2m               1m




     200N


                                                     400N

   For equilibrium to exists, torques must add to zero.
   The torques applied by the 2 kids are equal and opposite,
   so the see-saw doesn’t move.
Balanced torques happen on a seesaw when two children
  of different weights sit at different positions
We know that the heavier kid needs to sit
closer to the center of the see-saw. Why?

In order for the see-saw to balance, there
must be no rotation. This means that
there cannot be any unbalanced torques
acting on the see-saw.
The heavier kid has more weight (force) so
must have a smaller lever arm in order to
make the product of force and lever arm
(torque) equal that of the smaller kid sitting
further from the axis.
                      balanced torques
• Direction of rotation of applied torque is very important (i.e.
  clockwise or anticlockwise).
• Torques can add or oppose each other.
• If two opposing torques are of equal magnitude they will cancel one
  another to create a balanced system.
                            r1          r2



                   W1 = m1.g                        W2 = m2.g
                                 Torque = Fr

                                W1.r1 = W2.r2
                         or    m1.g.r1 = m2. g.r2

                   Thus at balance: m1.r1 = m2.r2
             (This is the principle of weighing scales.)
 If an object is supported at one
 special point, it will balance.
 This point is called the center of
 gravity .

•For SYMMETRICAL objects,
such as a meter stick, the CG is
located at the actual center.
• For ASYMMETRICAL objects,
such as a hammer, the CG is
located away from the center,
closer to the hammer head (the
heavier head)
               Center of Mass (CM)
• An object can be divided into
  many small particles
   – Each particle will have a
      specific mass and specific
      coordinates
• The x coordinate of the center
  of mass will be
                m x    i       i
        xCM    i

                m  i
                            i


• Similar expressions can be
  found for the y and z
                             A system can often be well represented
  coordinates
                              as though its mass were concentrated
                             at single point,
                             the center of mass
Center of Mass examples

           CM



      CM




           CM
CM for the Human Body


       The location of the center of mass
       of the leg (circled) will depend
       on the position of the leg.
High jumpers have developed a technique
where their CM actually passes under the bar
as they go over it. This allows them to clear
higher bars.




           CM
                  Center of Gravity
• All the various gravitational
  forces acting on all the
  various mass elements are
  equivalent to a single
  gravitational force acting
  through a single point called
  the center of gravity.



  The center of gravity is the point where the gravitational force can
  be considered to act. It is the same as the center of mass as long
  as the gravitational force does not vary among different parts of
  the object.
A force applied at the CG of an object results in straight-line
acceleration, but no rotation.
       A force applied away from the CG will have a “lever
arm”, which produces a torque. The result is straight-line
acceleration and rotation.
           Why things fall over
• Every object has a special point called the
  center of gravity (CG).
• if the center of gravity is supported, the object
  will not fall over.
Touch your toes
while standing against a wall




            Why things fall over

				
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posted:4/11/2011
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