Abstract:
The invention is an educational demonstration designed to help students better understand the concepts of torque and the inverse square law. A variable torque pulley consists of a rigid overlapping spiral shape member and a rigid circular member. The circular member is attached to the spiral shaped member as one integral piece, each incorporating a groove around the perimeter to act as a guide for a length of string. The circular member acts as a pulley and the spiral shape member acts as a lever. The variable torque pulley rotates in a circular fashion pivoting on an axle about its center. String is used with the circular member to rotate the apparatus. String is used with the spiral shape member for the hanging of a weight. The axle acts as the fulcrum for the spiral shape member. The shape of the spiral is important because as the variable torque lever rotates, the weight moves toward the fulcrum and the torque drops off in harmony with the inverse square law.

Description:
TECHNICAL FIELD 
     This invention relates to educational demonstrations, and more specifically to the apparatus of a variable torque pulley which is used to demonstrate the principles of torque and the inverse square law. 
     BACKGROUND OF THE INVENTION 
     Essentially the magnitude of all the forces at work, which we experience in our macroscopic world, vary with distance according to the inverse square law, such as the gravitational force, the magnetic force, and the electric force. Some examples of this are the following: The gravitational force between any two objects varies inversely with the square of their separation. The magnetic force between a magnet and a piece of iron varies inversely with the square of their separation. The electromagnetic force that binds atoms and molecules together is an electric force between charged elementary particles such as protons and electrons. This force varies as the inverse square of the separation of the particles. These concepts are abstract and difficult to understand, for it is not easy for most students to visualize this force distance relationship. This invention solves that problem by providing a physical demonstration that students can touch feel and experience for themselves to help them better understand the laws of the natural world. 
     SUMMARY OF THE INVENTION 
     In accordance with the present invention a variable torque pulley consists of a rigid overlapping spiral shape member and a rigid circular member. The circular member is attached to the spiral shaped member as one integral piece, each incorporating a groove around the perimeter to act as a guide for a length of string. The circular member acts as a pulley and the spiral shape member acts as a lever. The variable torque pulley rotates in a circular fashion pivoting on an axle about its center. String is used with the circular member to rotate the apparatus. String is also used with the spiral shape member for the hanging of a weight. The axle acts as the fulcrum for the spiral shape member. The shape of the spiral is important because as the variable torque pulley rotates, the weight moves toward the fulcrum and the torque drops off in harmony with the inverse square law. The invention helps students to better understand the concept of the inverse square law by providing a dynamic and tangible demonstration. The demonstration is interactive; such that it allows the student to feel the gravitational force of an object, in this case a weight, change with distance according to the inverse square law. Furthermore the invention allows students to visualize the concept of torque as a vector product consisting of both a force and a length of lever arm as the length of the arm changes. After finishing the experiment, the student can plot what he or she is visualizing and gain important practice with his or her graphing skills. 
    
    
     Other aspects of the present invention will become apparent from the following detailed description, taken in conjunction with the accompanying drawings, which illustrate, by way of example, the principles of the invention. 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a side elevation view of a variable torque pulley in accordance with the present invention; 
     FIG. 2 is an edge view of the variable torque pulley; 
     FIG. 3 is an opposite side elevation view of the variable torque pulley; 
     FIG. 4 is a perspective view of the variable torque pulley; 
     FIG. 5 is a reduced side elevation view of a system for demonstrating the inverse square law; 
     FIG. 6 is a reduced side elevation view of the system in a second position; 
     FIG. 7 is a reduced side elevation view of the system in a third position; 
     FIG. 8 is a reduced side elevation view of the system in a final position; 
     FIG. 9 is a first graph of force verses length; and, 
     FIG. 10 is a second graph of force verses length using a heaver weight. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     A preferred embodiment of the variable torque pulley  20  is illustrated in FIGS. 1 through 4. A variable torque pulley  20  consists of a spiral member  22  and a circular member  34 . The spiral member  22  is connected to the circular member  34  so that the first central axis  24  of spiral member  22  is coincident with the second central axis  36  of circular member  34  refer to FIG.  2 . Additional material is molded to the circular member  34  to act as a counter weight  42 . As the variable torque pulley rotates its center of gravity is always aligned with the central axes  24  and  36 . Pro engineering  3 D modeling software was used to design the variable torque pulley  20  and to facilitate the correct placement of the center of gravity. In other words the circular member  34  and counter weight  42  is attached to the spiral shaped member  22  all as one integral piece comprising the variable torque pulley  20 . The spiral member  22  contains a first line-receiving groove  28  along its first perimeter  26  and the circular member  34  contains a second line-receiving groove  40  around the second perimeter  38 . The purpose of the grooves is to provide a guide for the string. 
     The shape of the spiral member  22  must be derived from calculation to adhere to the inverse square law during operation. We will start with a simplified discussion of torque and move into the equations necessary for the construction of a variable torque pulley  20 . The magnitude of the torque is the vector product of the force Mg and the length of a lever arm D. Assume the line of action is perpendicular to the lever arm. Where T is the torque, M is the mass of the weight, g is the acceleration due to gravity and D is the length of the lever arm. This is the definition for torque. 
     
       
           T=MgD   [1] 
       
     
     Notice that D is directly proportional to T. A change in D will cause a corresponding change in T. Because we want the torque to drop off in harmony with the inverse square law we will change the length of the lever arm D. We want D to change with rotation as the inverse square of the angle of rotation θ with respect to a reference line  32 , so we write the equation.              D   =     1     θ   2               [   2   ]                                
     Next we substitute equation 2 into equation 1 and we get.              T   =     Mg     θ   2               [   3   ]                                
     Look back at equation 1 notice again that D is directly proportional to T. α is the symbol use to designate proportionality, so we write. 
     
       
           DαT   [4] 
       
     
     Substituting equation 3 into equation 4 we obtain.              D                 α                   Mg     θ   2               [   5   ]                                
     Because Mg is unknown at this time we will include it in an unknown constant we call C, now the equation becomes.              D   =     C     θ   2               [   6   ]                                
     When designing a variable torque pulley we must have a starting point and an ending point; consequently, we must choose a D initial and a D final. For D initial we use the equation.                D   i     =     C       (     θ   o     )     2               [   7   ]                                
     Where θ 0  is the starting angle. For D final we use the equation.                D   f     =     C       (       θ   o     +     2      n                 π       )     2               [   8   ]                                
     Where n is the number of turns in the spiral and the expression 2nπ is an angle in radians. Rearranging equation 6 we obtain an expression for C. 
     
       
           C=Di(θ   0 ) 2   [ 9 ] 
       
     
     Substituting equation 9 into equation 8 we obtain.          D   f     =           D   i          (     θ   o     )       2         (       θ   o     +     2      n                 π       )     2                              
     Next we solve the equation for θ 0 .                          D   f       D   i         =                  θ   o         θ   o     +     2      n                 π                       2      n                 π            D   f       D   i           =                  θ   o     -       θ   o              D   f       D   i                           2      n                 π            D   f       D   i           =                  θ   o          (     1   -         D   f       D   i           )                     θ   o     =                  2      n                 π            D   f       D   i             1   -         D   f       D   i                           [   10   ]                                
     Knowing the number of turns in the spiral n, the D i  and D f , we; can find the starting angle by using equation 10. Referring to FIG. 1, the general equation governing the shape of the spiral is as follows.              D   =     C       (       θ   o     +   θ     )     2               [   11   ]                                
     There are an infinite number of possibilities with regard to the size and configuration of the spiral member  22  and the circular member  34 . For example the radius of the circular member R in FIG. 1 can be chosen at any convenient size and the spiral member  22  can be designed to rotate around once, twice, three times, or ten times if desirable by changing the value of n. Df and Di can be chosen at any value. For example if D i  is chosen to be 12 inches and D f  is chosen to be 0.237 inches and the variable torque pulley  20  is to rotate around twice n=2 equation 10 is used to find θ 0  having a value of 2.05 radians. Equation 9 is used to find the constant C having a value of 50.66. The values of C and θ 0  are substituted into the general equation 11. D is found for various angles of θ, with respect to reference line  32 , in this case from 0 to 4π, producing the shape of the spiral member  22  seen in FIGS. 1 through 4. 
     The operation and function of the invention is shown by the system  100  for demonstrating the inverse square law illustrated in FIG. 5 through 10. The spiral member  22  contains a through-hole  31  at its distal end  30  for attachment of string. This section of string will be referred to as the first line  106 . The circular member  34  contains a though-hole  39  on its second perimeter  38  for the attachment of the second line  108 . A force-measuring device  110  such as a spring balance is attached to the end of the second line  108 . Any suitable weight  104 , also found in most college and high school stock rooms is attached to the end of the first line  106 . When selecting a weight, it is recommended not to choose one more then a few pounds, for too much weight will cause unnecessary stress to the bearings. Bearings are press fit into the front and back of the center hole of the variable torque pulley  20 . The bearings are of the flanged type to hold them in place. A rigid axle  102  is inserted into the bearings to allow rotation of the torque pulley  20 . The axle  102  is clamped to any suitable stand  112 , found in most college and high school stock rooms, so that axle  102  is disposed horizontally with respect to the support surface  700 . For optimum performance the stand  112  should be no shorter then four feet long. The second line  108  is attached to the circular member  34  by tying a knot on one end and threading it into the through hole  39  contained on the second line receiving groove  40 . Because n=2, this second line  108  is rapped twice around the inside of the second line receiving groove  40  contained in the circular member  34 . Conversely, One end of the first line  106  is tied in a similar fashion to the through-hole  31  at the distal end  30  of the spiral shape member  22  and the other end is tied to a weight  104 . The weight  104  rests on a suitable support surface  700  such as a tabletop or the floor of the classroom. The second line  108  is pulled in the direction of the arrow causing the variable torque pulley  20  to rotate. The first line  106  is guided by the first line-receiving groove  28  and the weight  104  is lifted upward. The point when the weight  104  just lifts off the support surface  700  is measured by the force-measuring device  110  as force initial F i  in FIG.  5 . The equations governing the operation and function of the invention are similar to the equations discussed above. The difference is that the angle of rotation θ 0  and θ is replaced with a length of string Lo and L and D i  and D f  are replaced with F i  and F f . R is the radius of the circle.                F   i     =     C       (     L   o     )     2               [   12   ]               C   =       (     F   i     )          L   o   2               [   13   ]                 F   f     =     C       (       L   o     +     2      n                 π                 R       )     2               [   14   ]                 L   o     =       2      n                 π                 R            F   f       F   i             1   -         F   f       F   i                     [   15   ]               F   =     C       (       L   o     +   L     )     2               [   16   ]                                
     Variable torque pulley  20  continues to rotate in a circular fashion pivoting on the axle  102  about its center. The axle  102  acts as the fulcrum for the spiral member  22 . The weight  104  moves toward the fulcrum effectively decreasing the length of the lever arm D. The force felt and recorded on the force measuring device  110  drops off in harmony with the inverse square law as the length L of the second line  108  increases. FIG. 6 illustrates the variable torque pulley  20  rotated clockwise 90 degrees or (π/2 radians) from its original position shown in FIG.  5 . FIG. 7 shows the rotation at 180 degrees (π radians) of rotation. FIG. 8 illustrates 720 degrees or (4π radians) of rotation at which point F f  is read from the force-measuring device  110 . FIG. 9 represents a graph where the force read from the force-measuring device  110  is plotted against the length (L o +L) of the second line  108  at 90-degree intervals. FIG. 10 is the same graph as FIG. 9 yet a heaver weight  104  is used. For example, referring to FIGS. 5,  6 ,  7 , and  8 , if Fi is read from the force measuring device to be 12 Newtons and Ff is read to be 0.2 Newtons and the radius of the circle R=2.5 inches, and n=2 then equation 15 can be used to calculate L o =4.66. Knowing L o  and F i  equation 13 can be used to calculate the constant C=260.59. The length of the second line  108  is measured with a ruler at each interval and recorded with its corresponding force read from the force-measuring device  110 . The graphs shown in FIGS. 9 and 10 are actual experimental results obtained by using this method with one of our prototypes. Table 1 refers to FIG.  9  and table 2 refers to FIG.  10 . 
     
       
         
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 L (inches) 
                 L + Lo (inches) 
                 F (newtons) 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Lo = 4.66″ 
                 0 
                 4.66 
                 12 
               
               
                   
                 Fi = 12(N) 
                 3.93 
                 8.59 
                 4.7 
               
               
                   
                 Ff = .2(N) 
                 7.85 
                 12.51 
                 1.9 
               
               
                   
                 C = 260.59 
                 11.78 
                 16.44 
                 1.2 
               
               
                   
                   
                 15.7 
                 20.36 
                 .8 
               
               
                   
                   
                 19.63 
                 24.29 
                 .5 
               
               
                   
                   
                 23.55 
                 28.21 
                 .39 
               
               
                   
                   
                 27.48 
                 32.14 
                 .3 
               
               
                   
                   
                 31.4 
                 36.06 
                 .2 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 L (inches) 
                 L + Lo (inches) 
                 F (newtons) 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Lo = 5.50″ 
                 0 
                 5.5 
                 18 
               
               
                   
                 Fi = 18(N) 
                 3.93 
                 9.43 
                 7 
               
               
                   
                 Ff = .4(N) 
                 7.85 
                 13.35 
                 3 
               
               
                   
                 C = 545.22 
                 11.78 
                 17.28 
                 1.8 
               
               
                   
                   
                 15.7 
                 21.2 
                 1.2 
               
               
                   
                   
                 19.63 
                 25.13 
                 .8 
               
               
                   
                   
                 23.55 
                 29.05 
                 .6 
               
               
                   
                   
                 27.48 
                 32.98 
                 .5 
               
               
                   
                   
                 31.4 
                 36.9 
                 .4 
               
               
                   
                   
               
             
          
         
       
     
     When comparing the experimental results with the expected results, it is important to take note of some possible sources of error. The force-measuring device  110  in this case a spring balance was used to achieve the results represented in FIG. 9 and 10. This spring balance is an economy model mechanical type spring balance found in most high school and college stock rooms. The balance was high in friction and low in precision. Although the experimental results are close to the expected results greater accuracy can be achieved by using a digital balance. The equation at the top of the graphs shown in FIGS. 9 and 10 is the equation of the curve representing the experimental results in the form shown below.        Y   =     C     X   2                              
     An excel spread sheet was used to find this equation. Y is the force and X is the length of the second line  108  L+L o . The student may now experience several physical laws once taught only on the chalkboard as a rather abstract principle. 
     In terms of use, a method for demonstrating the inverse square law, includes: 
     (a) providing a system  100  for demonstrating the inverse square law, including: 
     a variable torque pulley  20  for demonstrating the inverse square law, the variable torque pulley  20  including: 
     a spiral member  22  having a first central axis  24  and a first perimeter  26 , a first line-receiving groove  28  disposed around the first perimeter  28 , and a distal end  30 ; 
     the first perimeter  26  defined by an equation Dα1/θ 2 , where D is a distance from any point on the first perimeter  26  to the first central axis  24 , and θ 2  is the square of an angle θ formed by (1) a line between point D and the central axis  24 , and (2) a reference line  32 ; 
     a circular member  34  having a second central axis  36  and a second perimeter  38 , a second line-receiving groove  40  disposed around the second perimeter  38 ; 
     the spiral member  22  connected to the circular member  34  so that the first central axis  24  is coincident with the second central axis  36 ; 
     a counter weight  42  connected to the circular member  34 , so that a center of gravity of variable torque pulley  20  aligns with first  24  and second  36  central axes. 
     an axle  102  disposed along first  24  and second  36  central axes so that variable torque pulley  20  may rotate around axle  102 ; 
     a weight  104 ; 
     a first line  106  having a first end and a second end; 
     a second line  108  having a first end and a second end; 
     a force measuring device  110  having a first end and an opposite second end; 
     (b) connecting first end of first line  106  to distal end  30  of spiral member  22 ; 
     (c) connecting second end of first line  106  to weight  104 ; 
     (d) connecting first end of second line  108  to second perimeter  38  of circular member  34 ; 
     (e) connecting second end of second line  108  to said first end of force measuring device  110 ; 
     (f) placing second line  108  in second line-receiving groove  40  and wrapping second line  108  around second perimeter  40  of circular member  34 ; and, 
     (g) pulling second end of force measuring device  110 , thereby causing variable torque pulley  20  to rotate about axle  102 , and causing first line  106  to be received by first line-receiving groove  28  and wrap around first perimeter  26  of spiral member  22  and thereby lift weight  104 , and causing force measuring device  110  to measure a force. 
     The method further includes: 
     in step (a), first perimeter  26  extending around first central axis  24  for 720°. 
     The method further includes: 
     in step (a), D at distal end  30  being about 12 inches. 
     The method further including: 
     in step (a), circular member  34  having a radius R; and, 
     R being about 2.5 inches. 
     The method further including: 
     in step (a), a stand  112  having a bottom connected to axle  102 , so that the bottom of stand  112  may be placed upon a support surface  700 , thereby carrying variable torque pulley  20  so that axle  102  is disposed horizontally. 
     The method further including: 
     in step (a), wherein D=C/(θ 0 +θ) 2 , with C and θ 0 , being derived constants. 
     The method further including: 
     in step (g), the force measuring device  110  measuring a force according to the equation F=C/(L+Lo) 2 , wherein F is force, L is a length of pull, and C and Lo are derived constants. 
     The preferred embodiments of the invention described herein are exemplary and numerous modifications, dimensional variations, and rearrangements can be readily envisioned to achieve an equivalent result, all of which are intended to be embraced within the scope of the appended claims.