Worksheet
Lab Discussion: Gravitational Field Strength and the Acceleration Due to Gravity
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Method 1
Gravitational Fields
are regions surrounding a massive object in which other objects having mass will feel a gravitational force of attraction.
Gravitational fields are usually illustrated by using radial field vectors.
In this vector diagram, the gravitational field is stronger where the field lines are closer together.
Radial fields always obey an
inverse square relationship
.
This means that when you compare two positions in a radial field (A and B) where
R
_{B}
= 2R
_{A}
, the gravitational field at B will be (1/2)
^{2}
or 1/4
^{th}
as strong as it is at A.
In the above diagram, the central mass,
M
, is surrounded by a radial, gravitational field which means that the field lines look like extensions of the radius of a circle. All field lines point to the center of
M
. Since the three red masses,
m
, are all placed the same distance from the center of
M
they would feel the same magnitude force of gravitational attraction.
where G = 6.67 x 10
^{-11}
Nm
^{2}
/kg
^{2}
,
m
and
M
are measured in kilograms, and
r
is the radial distance from the center of
m
to the center of
M
measured in meters.
If we were to place a different mass at the same distance
r
from
M
(that is somewhere along the same dotted surface) if would feel a different magnitude of gravitational force.
If the unknown mass,
m
, was three times larger than
m
, then the gravitational force,
F
, would become
3F
. If the unknown mass was 10 times larger than
m
, then the gravitational force would become
10F
. That is, the gravitational force is directly proportional to the product of the two masses.
In our diagram, all of our masses are experiencing the same ratio of gravitational force to mass which is called the
gravitational field strength.
In our lab, we used a spring scale to measure the force of gravitational attraction between the Earth and the 8 different-sized balls. When purchased, the spring scale had been calibrated to recognize a given amount of weight (gravitational force) as a distinct stretch in the spring’s length.
Since the divisions on the spring scale were delineated in multiples of 0.4 newtons, our measurements were restricted to one decimal place and involved a large degree of estimation.
We then measured the gravitational mass of each ball using a triple beam balance. Although we could measure each mass up to three decimal places our ultimate ratio of the gravitational field strength, F/m, was limited to only one decimal place because of the restrictions on our measurement of each ball’s weight.
During one class period they obtained an average value of 11.24 N/kg, or 11.2 N/kg.
If we had used our formula and the values for the mass of the earth and the earth’s average radius, we would have obtained a value for the gravitational field strength at sea level to be
So the percent error for their calculation was about 14%.
Method 2
The second method of measurement to determine the strength of the earth’s gravitational field was to measure and record the slope of a velocity-time graph created by a motion detector tracking each ball’s descent.
When we examined the probe’s software we observed two graphs: one showing us
position vs time
and the second showing us
velocity vs time
.
In the equation to the left, the expression
net F
represents the sum of all the forces acting on an object. In our experiment, the only force acting on each object was the gravitational attraction between the ball and the earth. For air resistance to have had an impact on a ball's motion it would have had to have travelled much, much faster and have had a much larger surface area.
When the same group averaged all of its slopes from the 8 velocity-time graphs they got a value of 10.5 m/sec
^{2}
.
But what do the slopes of those velocity graphs represent? To answer this question we must define a new behavior called acceleration. Acceleration is the rate of change of velocity with respect to time. Recall that earlier in the year we defined inertia as a measure of the resistance an object has to being accelerated – or having its state of motion altered.
The equation
a = net F/m
is
Newton’s Second Equation of Motion
. This equation states that the acceleration an object experiences is directly proportional to the net force acting upon it and inversely proportional to the object’s mass. It represents a mathematical statement of what was happening to each released ball as it was being pulled downward by the attractive force of gravity.
Since this acceleration is so special it is called the
acceleration due to gravity
and is represented by the variable
g
. But notice that
g
is equal to the same expression as the gravitational field strength,
F/m
.
This relationship allows us to now calculate the percent error for the second part of the experiment again using 9.80 m/sec
^{2}
as the accepted answer. The error for their average slope was 7.1%.
SUMMARY
In summary, the average of our ratios of F/m and the average of our slopes for each velocity-time graph should have been the same, 9.80 m/sec
^{2}
. But be careful when working problems; this value changes with a planet’s mass and/or the object’s distance from the planet’s center.
Refer to the following information for the next eight questions.
Before we move away from this lab let’s practice some of the important equations and relationships.
(1) What is the gravitational force of attraction between two 5-kg masses held 1 meter apart?
(2) How would the attractive force in Question (1) change if the two masses were moved to new positions so that they are only separated by half a meter?
(3) How would the attractive force in Question (1) change if the two masses were moved to new positions located 10-meters apart?
(4) What does a 500-kg mass weigh on the surface of the earth?
(5) What does a 500-kg mass weigh in space if it is placed 60 earth radii above the surface of the earth?
(6) What would a 500-kg mass weigh on Mars if Mars has a mass approximately equal to 1/10th the mass of the earth and has a radius equal to roughly ½ of the radius of the earth?
(7) What is the acceleration due to gravity on the surface of Mars?
(8) Based on the following velocity-time graph obtained for a freely falling body on planet X, what is this planet’s acceleration due to gravity?
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