Flywheel experiment

               θ                    = total angle turned through during unwinding and rewinding (rads)

To calculate the angular acceleration, (α),

              S = u t + a t2/2                                                                  . . . . . . . .(2)

and       α =a/r                                                                                . . . . . . . . (3)

where, s             = distance travelled by mass during decent (m)

             u             = initial velocity of mass (=0)

             t              = time to travel distance s (s)

             a             = linear acceleration of mass (m/s2)

             r              = effective radius of the flywheel axle (m)

To determine, experimentally, the moment of inertia (Iexp);

     T – Tf = (I + m r2) α                    where T = m g r             . . . . . . . . . (4)

To calculate a theoretic value for I. The equation is;

     I = MR2/2                                                                              . . . . . . . . (5)

Where M            = mass of flywheel (kg)

              R            = radius of flywheel (m)

3. EXPERIMENTAL PROCEDURES

3.1   DESCRIPTION OF THE TEST EQUIPMENT

• Flywheel (disc and axle)
• Stopwatch
• String
• Ruler
• Pencil (to mark distance)
• Bracket
• Bearing
• Hook
• Mass (known) and mass holder
• Rope
• Board

                                 Flywheel

                         (disc + Axle)

         A        

             C          H1

                                 

        String

        H2

 

A known mass         B

Figure 3.1a

3.2   PROCEDURE

• The string is wrapped around the flywheel in a clockwise direction, which in turn lifts the known mass that is attached to the bottom of the string to a point close to the flywheel (point A on fig 3.1).
• The string, with the mass attached to it, is then allowed to wind down the flywheel until the mass reaches its lowest point (point B on fig 3.1), which is timed with a stop watch.
• The distance between points A and B is measured as H1.
• After reaching its lowest point, the mass then bounces back and starts to travel in the opposite direction, but then stops at a particular point (point C on fig 3.1).
• The distance between points B and C is measured as H2.
• The experiment is then repeated again, so as to improve reliability and accuracy of the supposed result.

3.3   RESULT                                                                                                        Table 3.3

H1 = Original height of mass after it unwinds from the flywheel

H2 = Final height of mass after bouncing back in opposite direction

 θ = Total angular displacement (rads)

r = Effective radius of the axle = 13.75x10-3m.

Radius of shaft and rope (r) = 0.01375m

Mass of flywheel = 6.859kg

Radius of flywheel = 0.1m

Radius of axle = 0.0125m

4.  ANALYSIS OF RESULTS

To calculate the moment of inertia of the flywheel;

             T – Tf = (I + m r2) α                    where T = m g r  

Make ‘I’ the subject of the formula;

            Iexp = (T – Tf )/α – (m r)

then, the value of T(applied torque) is;

T = m g r

   = 0.1 x 9.81 x (13.75x10-3)

   = 13.49x10-3 Nm

To calculate Tf (frictional torque);

Tf = mg (H1 – H2)/θ

    = (0.1 x 9.81 x 0.77)/ 56

    = 1.35x10-2 Nm

To calculate the angular acceleration (α);

α = 2H1/ (r x t2)

    = (2 x 0.98)/ (13.75x10-3 x 22.882)

    = 0.27ms-2

Therefore,

Iexp = (13.49x10-3 – 1.35x10-2)/0.27 - (0.1 x [13.75x10-3]2)

       = 3.7x10-5 – 1.89x10-5

       = 1.81x10-5 kgm2

To calculate the theoretic value for the moment of inertia;

Itheory = MR2/ 2

            = 6.859 x (0.1)2 / 2

            = 3.43 x 10-2 kgm2

% error = [(Expected Value – Actual value)/ Expected Value] x 100

               = [3.4x10-2/ 3.43x10-2] x 100 = 99.13%

5. DISCUSSIONS/CONCLUSION

Following the analysis of my results, the values of Iexperiment and Itheory differ by fairly a significant amount i.e. (a percentage error of 99.13%). The errors that led to the difference in the two values can be categorize into two sub-groups called “Measurement errors” and “Procedural errors”.

Measurement Errors.

• Errors may perhaps have crept up while measuring the distances of H1 and H2. These distances could have possibly been marked incorrect if the points were not marked at eye level, which could have lead to errors in the final value. However, these errors could have been minimised by taking more repeated readings, or even recording the experiment with the use of a video camera in order to help in checking for these kind of errors.
• Furthermore, another error that could have affected the final value was the timing of the stopwatch while measuring H1 and H2. This human error can be significantly reduced via total concentration of everyone involved in the experiment.

Procedural Errors.

The motion of the mass that was attached to the spring could have been affected by factors, such as the air resistance and friction, which would lead to easy energy loss during the experiment. This could have also led to some errors in the final value.                                                                                

This error could have been minimised by doing the experiment in a closed system, which would have not just minimised errors, but also increase the accuracy and reliability of the result.

Reference

1. Lynn White, Jr., “Theophilus Redivivus”, Technology and Culture, Vol. 5, No. 2. (Spring, 1964), Review, pp. 224-233 (233)1 
2.  , .
3.  Lynn White, Jr., “Medieval Engineering and the Sociology of Knowledge”, The Pacific Historical Review, Vol. 44, No. 1. (Feb., 1975), pp. 1-21 (6)