THIS Page provides updates and hints to the centripetal force lab; CLICK HERE for hints to each part. |
UPDATE #1. (Q-7) |F_{BEST} – m_{BEST·}·g| < 2(ΔF + Δmg). NOTE THAT THE EXPRESSION TO THE RIGHT OF THE OPERATOR MUST CONTAIN Δmg; there must be a g for gravitational force unit consistency. |
UPDATE #2 (Q-4) Because you are defining T from trials involving 25
rotations, then the "instrument" uncertainty is reduced by a factor
of 1/n = 1/25, where n = 25 is the number of trials . ΔT_{inst} = 0.001 s/ n or 0.0001 s / n, depending on its setting. In this experiment, the setting was for 0.0001 s. Thus, ΔT_{inst } = (0.0001/25) seconds = 0.000004 seconds. |
In this experiment, you test whether Newton’s
Second Law applied to uniform circular motion is correct. In particular you will confirm the validity
of the equation for the net force magnitude
F = Mv^{2}/r. The general mathematical background, subject to further class
discussion, is here: The force causing this special motion is called the centripetal force , or “center seeking” force. It points to the circle’s center and has constant magnitude F = mv^{2}/r, where v is the speed ( i.e. velocity magnitude) and r is the radius. It is not a new force, it is only a name we assign to any force that causes uniform circular motion, whether it be the gravitational and electrical force underlying the Earth’s orbit around the Sun and the electron’s motion around the nucleus, respectively. In the example of a rock twirled in a vertical circle, the centripetal force at any point is a combination of gravitational and tension forces. See chapter 5. To test the theory, you will measure the centripetal force magnitude F by two methods , static and dynamic: (A) A certain force F acting on the mass M is measured under static conditions by a weight mg. We could call this the static value of F. (B)
When the mass under goes uniform circular motion this force should be equal to the
centripetal force, and thus equal to Mv^{2}/r .
Thus we are testing whether the static and dynamic values are equal : The experiment can easily be illustrated with a free body
diagram provided by the instructor-- See figure 1, page 3, of the experimental set up described here: (A) STATIC MEASUREMENT. Make
sure the axle-to-bob spring and the bob-to-pulley string are precisely horizontal by measuring the
height above the table at both ends, and then raising and lowering the pulley
until the heights are equal. You can
make the platform parallel with the table by adjusting the clamps and knobs. Also measure r, the distance between the axis of rotation to the center of the bob. Take 5 readings. Again, we can state the precision of our readings by assuming only an “instrument” uncertainty due to random errors from just reading the instrument. We will assume that the accuracy of any single reading with a meter stick is plus or minus ¼ of the smallest scale division of 0.1 cm. See page 36 here: http://dept.physics.upenn.edu/~uglabs/lab_manual/Lab_Manual.pdf Thus , Δr_{inst} = 0.1/4 = 0. 025 = 0.03 for a single reading
with a meter stick . But the length r
is measured from a difference of labeled marks on the stick. It can be shown that a more accurate “instrument”
uncertainty is times the
uncertainty for a single reading. See pages 49-50 here: http://dept.physics.upenn.edu/~uglabs/lab_manual/Lab_Manual.pdf (B)
DYNAMIC MEASUREMENT . To measure the dynamic value of
the centripetal force, you need to measure
Mv^{2}/r . This is done by measuring r (see Q-1) and the period T. (Q- 3) Measure M: Again, we can state the precision of our readings by assuming only an “instrument” uncertainty due to random errors from just reading the instrument. We will assume that the accuracy of any single reading with a mass scale is plus or minus ¼ of the smallest scale division of 0.1 g. See page 36 here: http://dept.physics.upenn.edu/~uglabs/lab_manual/Lab_Manual.pdf Thus , ΔM_{inst} = 0.1/4 = 0. 025 = 0.03 for a single reading with a mass scale. But the measurement may be “non-simple . We thus make the M measurement taking 5 readings. In this case, we state the precision by taking the difference between the minimum and maximum values and dividing by N (where N = 5.) We call this quantity R/N. Compute R/N . Compute the average (M_{BEST}) of the 5 measured masses. The uncertainty you choose will be either ΔM _{inst} or R/N , which ever is larger. In either case, the uncertainty is rounded to just one digit and we round the average (“best”) value to the same decimal place. The result is reported as M_{BEST} ± uncertainty .
(Q-5) Using (Q–1), (Q-3 ) and (Q-4), compute the dynamic value of F_{BEST} using equation (3). For the value of r , M and T use the “best” value computed in (Q–1), (Q-3 ) and (Q-4) (Q-6) . Beginning
with equations (3), it is easy
to derive with Math 1 and 3 calculus (Q-7) Apply the following test to determine
whether the data from this experiment supports the |