Discuss the below:
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.
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
g = (4πr2) / 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!)
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r (meters)
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10 T (seconds)
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T (seconds)
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g (m/s2)
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III. Average your values of g: _________________ m/s2
IV. Determine your error (take the more accurate value of g to be 9.80 m/s2).