M-4  Acceleration in Free Fall

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OBJECTIVE: To measure ``g'', the acceleration of gravity.

APPARATUS:

Free fall equipment: cylindrical bobs (identical except in mass) which attach to paper tape for recording spark positions; spark timer giving sparks every 1/60 s; cushion; non-streamline bobs to study air resistance (by adding a front plate).

INTRODUCTION:

As shown in figure below the spark timer causes sparks to jump from sharp point A (flush with the convex surface of the plastic insulator.) through the falling vertical paper tape to the opposite sharp point B flush with the other insulator surface. As the bob (plus paper tape) undergoes free fall, sparks from A to B mark the paper tape's position every 1/60 $s$. These data give the bob's acceleration in free fall, $g$, if air resistance is negligible. We'll think about air resistance later.

EXPERIMENTAL SUGGESTIONS:

1. Position spark timer chassis near table edge and with the convex surfaces (A and B) extended beyond table edge. Put cushion on floor directly under A and B.

2. Select the heaviest bob (no front plate). Insert one end of about a meter of paper tape between the two halves of the cylindrical bob and fasten together with thumb screw.

3. Insert paper tape between A and B. Hold the tape end high enough (vertically) above A and B that the bob just touches below A and B (and is centered). Start the spark and immediately release tape plus bob. Discard any part of tape which fell through the spark gap after bob hit the cushion.

4.

Fasten sparked tape to table top with masking tape. Place a meter stick on its side (on top of tape) so that ends of the mm graduations touch the dot track on the tape; this avoids parallax error (Appendix 4.)

5.

Ignore the first spark dot; then mark and measure the position of every other of the first 24 dots, thus using 1/30 s as the time interval instead of 1/60 s.
Estimate the dot positions to 0.1 mm, and assume this is your uncertainty, $\delta_r = 0.1$mm. Don't move meter stick between readings! Tabulate as in sample below.

6.

Check the measurement set by remeasuring the tape after moving the meter stick so that the recorded dot positions will be different. The differences should be the same. How close are they?

COMPUTER GENERATED TABULAR FORM FOR DATA:

Spark Real Position of Average Average

interval time every 2nd velocity acceleration spark (dot)

$$i t(i) \overline{r}(i) \pm \Delta r \overline{v}(i) \pm \Delta v \overline{a}(i) \pm \Delta a$$

$$\ldots \ldots$$ Units sec mm or cm

$$\delta_t = \delta_r = \delta_v = \delta_a =$$ Uncertainty

$$t(0) a$$   0 -- --

$$t(1) b b-a (c-b)-(b-a)$$   1

$$t(2) c c-b (d-c)-(c-b)$$   2

$$t(3) d d-c (e-d)-(d-c)$$   3

$$t(4) e e-d (f-e)-(e-d)$$   4

$$t(5) f f-e (g-f)-(f-e)$$   5

$$t(6) g g-f (h-g)-(g-f)$$   6

$$t(7) h h-g (i-h)-(h-g)$$   7

$$t(8) i i-h (j-i)-(i-h)$$   8

$$t(9) j j-i (k-j)-(j-i)$$   9

$$t(10) k k-j (l-k)-(k-j)$$ 10

$$t(11) l l-k (m-l)-(l-k)$$ 11

$$\vdots \vdots \vdots \vdots \vdots$$

SUGGESTIONS ON HANDLING DATA:

1. Let $t=0$ be where your readings start. Tabulate the actual readings, $r$, on the meter stick at the end of each time interval (as in column 3 above); calculate the average velocity, $\overline{v}$ in each interval; then calculate the average acceleration, $\overline{a}$ during each interval. If you do the table by hand it may be easier to compute everything in per time interval units and scale your answer when everything is done.
2. Find the average of the average $\overline{a}$ in each interval and convert (if necessary). This average of the average $\overline{a}$, $\langle \overline{a} \rangle$'' is much more accurate than fluctuations in the $\overline{a}$ column might indicate. Explain why. (Hint: Are the values independent? See last part, Suggestion 4) below.
3. With 1/30 s as the time interval, plot ``$r$'' vs time, $\overline{v}$ vs time, and $\overline{a}$ vs time. Note that for $\overline{v}$ the value is the average of v between to adjacent rows, $i$ and $i+1$, and therefore equals the instantaneous v at the middle of the interval.
4. Use the slope of the velocity curve to find acceleration. Your can do this by hand but graphical analysis tools should be available. This approach is a better way of using all the data than the above numerical one since averaging the $\overline{a}$ column involves summing the $\overline{a}$ column and in such a sum all readings except $(b-a)$ and $(m-l)$ drop out!

ERROR ANALYSIS:

If readings $a$, $b$, $c$, etc., are good to $\approx\frac{1}{2}$ mm because of irregularity in the spark path, the worst case would be for $(b-a)$ to be in error by 1 mm. Since these two measurements are to first order, uncorrelated, the errors should add in quadrature [see section on Errors]. Using every $v(i)$ for your slope determination is problematic because each $v(i)$ and $v(i+1)$ is correlated. This means that if $v(1) \propto b-a$ then $v(2) \propto c-b$, so that a fluctuation in $b$ will always push the adjacent values of $v$ apart. Thus a linear-fit of slope suffers from the same shortcoming as the average of the average acceleration calculation noted in Suggestion 4! Calculate the slope using only every other $v(i)$ because $v(1)$, $v(3)$, $v(5)$ etc. use independent data. Does your new value for $g$ vary much? Estimate the uncertainty in the slope. Spark timing errors are negligible. Air resistance is a systematic error, small for the streamline bobs at low velocities, but see the optional experiment below.

EQUIVALENCE OF GRAVITATIONAL AND INERTIAL MASS:

Galileo showed (crudely) that the acceleration of falling bodies was independent of the mass. Use the light plastic bob (identical in size and shape to the massive bob) to repeat the free fall experiment and thus check quantitatively this equivalence (when air resistance effects are small enough). If you are planning to do the optional experiment which follows, you should skip this part.

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LOCAL VALUE OF $g$: The UW Geophysics Department determined $g$ accurately for room 4300 Sterling Hall. A plaque on the northwest window sill gives $g = 9.803636 + 0.000001$ m/s$^2$.

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QUESTION: Do your value(s) of `` $g$ '' agree within your assigned errors?

OPTIONAL EXPERIMENTS

A.

EFFECT OF AIR RESISTANCE: While air effects are small for streamlined objects at low velocities, they can become large, e.g. on a parachute. To observe and correct for them, first make the bobs non-streamline by inserting into the bottom of the bob the banana plug holding a small flat plate. Then repeat the experiment for the non-streamline bobs (front plate attached), identical except for mass. There are four possible mass combinations. Since the force of air resistance, $f(v)$, is a function of only velocity if the size, shape and roughness are the same, then $f(v)$ on the two bobs will be almost the same since their velocities are similar. The net force on the falling body is then $F=mg-f(v)$. Hence

$$a = \frac{F}{m} = \frac{mg -f(v)}{m} = g - \frac{f(v)}{m}$$

and

$$\langle a \rangle \approx g - \left(\frac{1}{m}\right) \langle f(v) \rangle   .$$

Thus if we measure $\langle a \rangle$ for bobs identical except in mass and plot $\langle a \rangle$ against $1/m$, we should obtain a straight line whose extrapolation to $(1/m) = 0$ should give $g$. To further test the validity of this hypothesis you should plot the same data as recorded by other lab groups on your plot. If the data permits a simple linear fit you should be able extrapolate to infinite mass (i.e. the $y$-intercept).

B.

MEASUREMENT OF REACTION TIME BY FREE FALL:

(1)

With thumb and forefinger grasp a vertical meter stick at the 50 cm mark. Release and grab it again as quickly as possible. From the distance through which the 50 cm mark fell, calculate the time of free fall of the meter stick. This time is your total reaction time involved in releasing and grasping again.

(2)

Have your partner hold the vertical meter stick while you place your thumb and forefinger opposite the 50 cm mark but not grasping it. When your partner releases the stick, grab it as soon as possible. Again from the distance through which the 50 cm mark fell, calculate the time of free fall of the meter stick. Compare this time with the other method. Why may these reaction times be different?

OPTIONAL QUESTIONS:

1. Estimate the effect on your $g$ value of the air's buoyant force, $F_{a}$ for air density, $\rho=1.2$ kg/m$^{3}$ and brass bob density, $\rho_{\mbox{\small bob}}=8700$ kg/m$^{3}$.
   Hint: $F_{a}=\rho_{a}m_{\mbox{\small bob}}g/\rho_{\mbox{\small bob}}$    (why?), and $a =g[1-(\rho_{a}/\rho_{\mbox{\small bob}})]$    (why?).

2. According to universal gravitation, the moon also accelerates the bob with a value $a_{m}$ of

   $a_{m}=\frac{GM_{m}}{r^{2}} =\frac{6.67\times 10^{-11}(7.34\times 10^{22})}{(3.84\times 10^{8})^{2}} =0.000033 m/s^{2}$

   where $G$ is the constant of universal gravitation, $M_{m}$ is the mass of the moon, and $r$ is the distance between the moon and the bob. Since this $a_{m}$ is 33 times the uncertainty quoted for the local $g$ value, why doesn't the plaque also indicate the position of the moon at the time of measurement?

   Hint: Remember the acceleration $g$ in an orbiting earth satellite provides the centripetal acceleration for the circular motion but does not appear as ``weight" of an object in the satellite. While to first order the moon and sun effects are negligible, there are detectable tidal effects in the earth $(\approx 10^{-7} g)$ which one corrects for in the absolute measurements. See Handbuch der Physics, Vol XLVIII, p. 811; also Wollard and Rose, ``International Gravity Measurements", UW Geophysical and Polar Research Center, (1963) p. 183. On pages 211 and 236, they also describe the accurate absolute $g$ determination with two quartz pendula in room 70 Science Hall from which the plaque value in Room 4300 Sterling derives.