Objective To measure the centripetal force for whirling a mass round a horizontal circle and compare the result with the theoretical value given by F= mw2r .

The string may not be horizontal as the rubber bung moves around. In fact, the bung moves in a circle of radius r = L sinθ.

The tension T thus provides both the centripetal force and a force to support the weight of the bung. By resolving T into its horizontal and the vertical components, it is easy to show that T = mω2L regardless of the angle θ.

Procedure

1. Construct the centripetal force apparatus as shown in the following figure.

2. Find the mass of the rubber bung and the screw nuts. The weight of the screw nuts gives the tension T in the string.
3. Measure a length of the nylon string from the rubber bung to the glass tube. Mark the length L of the string with the paper marker. Record L.
4. Hold the glass tube vertically and whirl the rubber bung around so that the paper marker is just below the glass tube without touching it.

5. Time 50 revolutions of the bung and calculate the angular velocity ω.
6. Repeat several times using different lengths L of the string.

Result

Tabulate the results as follows:

        Mass of rubber bung m =   0.03491   kg

        Mass of screw nuts M =   0.13   kg

        ⇒ Tension in string T = Mg =   0.13   × 9.8 N =   1.274   N

        Mean mω2L =   1.264   N

Conclusion

Form the results, we can find that the length of the string L is increasing, the value of angular velocity ω is decreasing. The length of the string is indirectly proportional to the value of angular velocity. This is in accordance with the equation, ω=v/r .

Also, the value of mω2L is 0.785% smaller than the value of tension in string. They are almost equal.

Discussion

T = mω2L regardless of the angleθ

The tension, T in the string is provided by the weight of screw nuts, Mg.

i.e. T = Mg

The length, L of the string above the top of glass tube is between the distance of the paper marker and the bottom of the tube.

For vertical equilibrium of the rubber bung:

T cosθ = mg

Mg cosθ = mg

cosθ = m/M

∴θ = constant

For horizontal circular motion:

r = L sinθ

T sinθ = mω2L sinθ

T  = mω2L

∴T = mω2L regardless of the angleθ

Measured value of the tension T and the theoretical value mω2L almost equal

Measured value of the tension T=1.264N

Theoretical value mω2L = 1.274N

The measured value of the tension T is 0.785% less than the theoretical value mω2L .

They are almost equal.

Possible sources of errors

1. Friction exists between the glass tube and the string.
2. The rubber bung is not set into a horizontal circular path.
3. The rubber bung does not move with constant speed.
4. The length of the string beyond the upper opening is not constant.

θ increases with ω

Vertical components is T cosθ

Horizontal components is T sinθ

The system has no vertical acceleration

∴T cosθ = mg

The horizontal component of tension provides the centripetal acceleration

∴T sinθ = mrω2

Let L be the length of the string

i.e.  r = L sinθ

T sinθ = mrω2

T sinθ = m(L sinθ)ω2

T = mLω2

 mg/cosθ= mLω2

θ increases with ω

∴When the rubber bung is whirled around with a higher angular velocity ω , the angleθ becomes larger.

Reference

1. Level practical physics for TAS p. 28 - 30
2. Physics Beyond 2000 p. 40
3. http://en.wikipedia.org/wiki/Centripetal_force
4. http://www.greenandwhite.net/~chbut/centripetal_force.htm