Comparing Velocity of Objects Based on Mass; Pendulum

The Acceleration Due to Gravity

Background:

Isaac Newton investigated the nature of gravity in holding stellar and
planetary systems together. He devised a theory and a method to
determine, with great accuracy, the motion of planets due to their
mutual attraction and that of the sun. His theory is incorporated into
his “Law of Gravity” and his “Laws of Motion.”

Newton’s Laws predict that all bodies, left to fall near the surface
of earth, do so with the same change of velocity (called acceleration),
no matter what their mass.

Near the surface of the earth, free-falling bodies fall at an
acceleration of about 9.80 meters per second per second.

With a very good clock and length measure, the value of “g”, the
acceleration due to gravity, can be found to five or six decimal places.
This is precise enough to detect the difference in height you might be
above sea level. The value of “g” is not a constant, but diminishes
as you go away from the planet. Newton predicted that g=GM / r2 ,
where G is a universal constant (6.67x10-11Newton-meter2/kilogram2), M
is the mass of the planet, and r the distance the body is from the
center of the planet.

Newton’s Laws give that without air friction, that all bodies fall,
near the earth, an increasing distance with each interval of time,
according to: y=(1/2)g t2, where y=distance fallen, g=acceleration, and
t=time of fall. The effects of atmospheric friction are not large for
dense bodies when the body has not yet reached a relatively fast speed.

Equipment:

Clock or watch with display of seconds, a nickel, a metal paper clip, a
uniformly dense ball, and a piece of flat paper or card cut to a
3”x3” size, a flat table which can be tilted, a ruler (with
centimeters).

Procedure:

You can crudely check Newton’s predictions for yourself.

I. Simultaneously drop the paperclip and a dense ball from a height of
two meters. Do they hit the floor or ground at the same time? How
about the paperclip and the flat paper?

II. Explain.

III. Again from 2.0 meters, estimate the time to fall for a dense ball.
(As this time is predicted to be short, your estimate will be crude
without an accurate timer.) Make your estimate as a range of
possibilities (such as greater than .xxx. and less than .xxx). From
g=2h/t2, figure out the range of “g” you observed.

We can “slow down” the acceleration of these bodies by having them
roll down an incline. (However, this adds complications besides air
friction.)

IV. Release the coin or a uniformly dense ball from rest at the top end
of a table tilted to about 5 degrees, estimating time to reach the lower
end of the table. Measure the length along the table and the height of
the top end above the bottom end.

If friction can be neglected, Newton’s Laws predict the time to roll
as t=L  [2(1+k)/g h]1/2, where L is the length along the table, k is
1/2 for the nickel, 2/5 for the ball, g =is the acceleration due to
gravity, and h is the height of the high end of the table.

Practice this technique a few times. For the next four times, estimate
the time to roll for the nickel and then for the ball.

Coin Ball















Average these times:





V. How much error do you expect in your estimate of the time of roll?

The acceleration, using the formula above, should be g=2(1+k)
L2 / [h t2]

Use this formula to calculate a range of g with your average time of
roll minus you estimated error, and then with your highest estimated
roll plus your estimated error in time.

A more accurate way to easily calculate the acceleration due to gravity
is to use the period of a simple pendulum. The period is the time it
takes for the pendulum to swing back and forth once. It depends only on
the length of the pendulum and the acceleration due to gravity. We can
perform an experiment using a home-made pendulum to find the
gravitational acceleration to a fair degree of accuracy, using the
formula

π2r) / T2

Where r is the length of the pendulum, g is the acceleration due to
gravity, and T is the period of the pendulum.

Equipment:

A length of string, a compact object which can be tied to the string
like a metal washer, a tack, a stopwatch or clock, and a meter or yard
stick.

Procedure:

I. Tie your object to the string and tie or tack the other end somewhere
it can swing freely, such as a shower rod or underneath a table. The
string and hanging object should not drag or rub against anything.
Measure in meters (or convert from inches) the distance from the
approximate center of the object to the tack or knot.

II. Pull the object back so that it makes a small angle with the
vertical (about 20 degrees or less). Leaving your hand where it is,
release the object and let it make ten complete swings (forward and back
is one swing). Start the stopwatch when you release the object and stop
it when it returns for the tenth time. Record the time. Repeat the
exercise four more times, and use your value of r and values of T to
calculate g. (Note: The formula only works for small angles!)

r (meters) 10 T (seconds) T (seconds) g (m/s2)























III. Average your values of g: _________________ m/s2

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