Statement of Purpose: Estimate the total amount of matter which makes up the earth (mass) using a pendulum of one meter under the influence of gravity.
Statement of Procedure: I created a pendulum of one meter (measured from center of weight (or “bob”) to anchor point). Using this pendulum I calculated the acceleration of gravity by timing a 15° angle of swing and finding an average time per swing (out of 10 swings). Taking the (rounded) average time per swing (T) and the length of my pendulum (1 meter) (L), I used the equation T=2π√ (L/g) to find the acceleration of gravity (g) in m/s2. Then using the found acceleration of gravity (g), the known Universal Gravitational Constant (6.673 x 10-11 m3 kg-1 s-2) (G) and the known radius of the Earth (6378km or 6378000m or 6.378 x 106m) (R) I found the mass of the Earth in kilograms (M) using the equation M=g(R2/G). I repeated this ten times to average my results for the mass of the Earth.
Data:
Trial
|
Length in Meters (L)
|
Time of 10 swings in seconds
|
Time of 10 swings in seconds (rounded)(t)
|
Average Time per swing in seconds
t/10 (T)
|
Acceleration of Gravity in m/s2 (g)
|
Mass of Earth in Kilograms
|
1
|
1
|
19.85
|
20
|
2
|
10
|
6.02 x 1024
|
2
|
1
|
19.66
|
19.5
|
1.9
|
11
|
6.67 x 1024
|
3
|
1
|
20.10
|
20
|
2
|
10
|
6.02 x 1024
|
4
|
1
|
19.93
|
20
|
2
|
10
|
6.02 x 1024
|
5
|
1
|
19.89
|
20
|
2
|
10
|
6.02 x 1024
|
6
|
1
|
20.21
|
20
|
2
|
10
|
6.02 x 1024
|
7
|
1
|
20.24
|
20
|
2
|
10
|
6.02 x 1024
|
8
|
1
|
20.02
|
20
|
2
|
10
|
6.02 x 1024
|
9
|
1
|
20.38
|
20
|
2
|
10
|
6.02 x 1024
|
10
|
1
|
20.26
|
20
|
2
|
10
|
6.02 x 1024
|
Average Mass =
|
6.08 x 1024
|
Statement of Conclusion: My final conclusion is that the Earth has a mass of approximately 6.08 x 1024kg. After looking up the actual mass of the Earth (5.9722 x 1024) I am fairly pleased with my estimation. I was estimating 15°, my measured string may not have been precisely one meter and my stop watch times were probably a little off due to response time, plus I rounded them anyways. If I had more accurately measured 15°, somehow measured exactly one meter of string and somehow automated an instantly reacting stop watch and used those accurate measurements, my final calculations would probably be much closer to the known acceleration of gravity on Earth (9.807m/s2) and mass of the Earth. Also, using Newton’s Law of Conservation of Angular Momentum (angular momentum(g) = mass x velocity(T) x radius(L)) I was able to find that the mass of my bob was about 5kg, which I had not previously measured but would estimate is accurate.
1) What is the effect of the length of the string on the period of a pendulum? That is, how do the periods compare for pendulums of different length? Would pendulums of different length give the same or different values for the mass of the Earth?
a. Newton’s Law of Conservation of Angular Momentum (angular momentum = mvr) says that as the length (or radius) of an orbit increases, the mass or velocity must decrease to maintain angular momentum. Since the mass of the pendulum bob cannot decrease, the velocity must. So a longer string on the pendulum (radius) must result in a longer period. The opposite would be true if the pendulum string were shortened, the swing period would be shorter in order to maintain its constant angular momentum.
b. In this activity pendulums of different length should have the same results for the mass of the Earth. The equation to find the acceleration of gravity takes the length (L) and the average time per swing (T) into consideration T=2π√ (L/g). As the length of the pendulum increases, the time should increase proportionately. Also, the law of conservation of angular momentum (see “a”) states that length is proportional to velocity but not mass.
2) Is the strength of gravity the same everywhere? Put another way, does the acceleration of gravity – the rate at which things fall – different on different planets? Why or why not?
a. According to the Universal Law of Gravitation (Fg=GM1M2/R2), the strength of gravity(Fg) weakens with distance(R) in a squarely inverse proportion (2x distance = ¼ gravitational force) between two objects. We feel the same pull of gravity because we (average human) generally do not travel a distance far enough from the center (or surface) of the Earth to feel a noticeable change. So the strength of gravity is the same everywhere on the surface of the Earth, but it does change when there is a notable distance between two objects.
b. The acceleration of gravity(Fg) will change on another planet that does not have the same mass and radius as Earth. While all objects (regardless of mass) will accelerate at the same rate (providing no other forces act on it) no matter which planet you are on, the rate of acceleration will be determined by the radius and mass of the planet.
c. The Gravitational Constant (G) is the attraction between two objects and it is the same everywhere.
3) Would the period of a one meter pendulum be the same on Earth and the Moon? Why or why not?
a. Because the Moon is less massive (7.36 x 1022kg) this experiment would have very different results. The lesser mass of the Moon creates a much weaker gravitational pull (1.63m/s2), so a pendulum of one meter would swing (accelerate) much slower on the Moon than it does on the Earth. Using the same formula as before (T=2π√ (L/g)) but plugging in the moon’s gravitational acceleration, the same experiment would have an approximate swing period (T) of about 5 seconds on the moon as opposed to the swing period (T) of about 2 seconds on the Earth.
4) Why do you think we refer to the Universal Law of Gravity and the Universal gravitational Constant? Why “Universal”?
a. Gravity is everywhere. It may lose strength over distance, but every object which has mass has gravity and is thusly attracted to and attracts other objects of mass. This is what we refer to as the Universal Law of Gravity. This law does not only apply on Earth or in our solar system, it is a Universal Law that applies to all massive objects in the universe.
The Universal Gravitational Constant (big G) is the measure of the attraction between any two (or more) massive objects. This attraction is constant throughout the universe, however depending on the mass and distance of the objects will determine the acceleration due to gravity (small g).
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