Newton’s Law of Universal Gravitation
Newton’s Law of Universal Gravitation
Newton found that the gravitational force of
attraction |Fg| between two masses is:
directly proportional to the product of the
two masses, so |Fg| ∝ (m1)(m2)
inversely proportional to the square of the
distance between their centers, so
|Fg| ∝ 1/r2
By Newton’s 3rd law, the gravitational force
acts mutually on both objects, not just one!
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
Specific Outcome:
i. I can describe, qualitatively and quantitatively, Newton’s law of universal
gravitation.
ii. I can explain, qualitatively, the principles pertinent to the Cavendish
experiment used to determine the universal gravitational constant, G.
Newton’s Law of Universal Gravitation
The Cavendish Experiment
Newton’s Law of Universal Gravitation
Newton’s Law of Universal Gravitation
Combining these two relationships, we get:
|Fg| ∝
m1m2
r2
The proportional sign (∝) doesn’t mean
“equal,” so this is not an actual equation
If a constant is added, we have an equation
Later, we will look at how the universal
gravitational constant (G) was determined
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
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Newton’s Law of Universal Gravitation
Newton’s Law of Universal Gravitation:
where:
Gm1m2
Fg =
r2
Fg = gravitational force (N or
kg●m/s2)
m1, m2 = object masses (kg)
G = universal gravitational
constant (6.67 x 10-11 N●m2/kg2)
Newton’s Law of Universal Gravitation
ex. What is the force of attraction between a 20 kg
mass and a 50 kg mass separated by 50 cm?
r = 50 cm /100 = 0.50 m
Fg =
Gm1m2
r2
=
(6.67 x 10-11)(20 kg)(50 kg)
(0.50 m)2
= 2.7 x 10-7 N
r = center-to-center distance (m)
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
Newton’s Law of Universal Gravitation
ex. A person has a mass of 60 kg.
a)What is the gravitational force of attraction
between the person and the Earth?
Fg =
=
Gm1m2
r2
(6.67 x 10-11)(60 kg)(5.97 x 1024 kg)
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
Newton’s Law of Universal Gravitation
ex. A person has a mass of 60 kg.
b) What is the magnitude of his weight on the
surface of the Earth?
Fg = |mg| = |(60 kg)(-9.81 m/s2)|
= 5.9 x 102 N
(6.37 x 106 m)2
= 5.9 x 102 N
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
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Newton’s Law of Universal Gravitation
ex. There is a gravitational force of 10 N between two
objects. If both masses are tripled and the distance
between them is doubled, what is the new force?
BEFORE: Fg =
AFTER:
Fg =
=
Gm1m2
r2
(2r)2
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ex. There is a gravitational force of 10 N between two
objects. If both masses are tripled and the distance
between them is doubled, what is the new force?
ALTERNATE
APPROACH:
= 10 N
G(3m1)(3m2)
Newton’s Law of Universal Gravitation
=
9Gm1m2
4r2
Fg =
Newton’s Law of Universal Gravitation
ex. The gravitational force between two boxes is
3.2 x 10-7 N. One mass is tripled and the other mass
is halved. The distance is now one tenth of its
original value. Determine the new force.
AFTER:
Fg =
=
r2
Effect on Fg
10 N
Triple m1
30 N
Triple m2
90 N
Double r
23 N
(10 N) = 23 N
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
BEFORE: Fg =
Gm1m2
Change made
ORIGINAL
Gm1m2
r2
= 3.2 x 10-7 N
G(3m1)(0.5m2)
(0.1r)2
(3)(0.5)(3.2 x
=
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
Newton’s Law of Universal Gravitation
ex. The gravitational force between two boxes is
3.2 x 10-7 N. One mass is tripled and the other mass
is halved. The distance is now one tenth of its
original value. Determine the new force.
ALTERNATE
APPROACH:
(3)(0.5)Gm1m2
10-7
(0.1)2r2
N)
= 4.8 x 10-5 N
(0.1)2
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
Fg =
Gm1m2
r2
Change made
ORIGINAL
Effect on Fg
3.2 x 10-7 N
Triple m1
Halve m2
r/10
9.6 x 10-7 N
4.8 x 10-7 N
4.8 x 10-5 N
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
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Newton’s Law of Universal Gravitation
ex. Two objects have a
BEFORE:
gravitational force of 1.4 N
Gm1m2
= 1.4 N
between them. Determine Fg =
r2
the new force, given the
AFTER:
following diagrams.
BEFORE
m1
Fg =
m2
r
AFTER
m1 m1
m2 m2
m2
m1 m1
3r
=
G(4m1)(3m2)
(3r)2
12Gm1m2
9r2
= 1.9 N
Newton’s Law of Universal Gravitation
ex. Two objects have a
ALTERNATE APPROACH:
gravitational force of 1.4 N
Gm1m2
between them. Determine
Fg =
r2
the new force, given the
following diagrams.
Change made Effect on F
m1
=
12
9
(1.4 N)
m2
r
AFTER
m1 m1
m2 m2
m2
m1 m1
Newton’s Law of Universal Gravitation
ex. Two objects experience
BEFORE:
a gravitational force of
Gm1m2
= 135 N
135 N. Determine the
Fg =
r2
new force, given the
AFTER:
following diagrams.
10 kg
30 kg
r
AFTER
20 kg
30 kg
r/5
Fg =
G(2m1)(m2)
ORIGINAL
1.4 N
Quadruple m1
5.6 N
Triple m2
16.8 N
Triple r
1.9 N
3r
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
BEFORE
g
BEFORE
2Gm1m2
=
(0.2r)2
(0.2)2r2
2
=
(135 N)
(0.2)2
= 6.75 x 103 N
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
Newton’s Law of Universal Gravitation
ex. Two objects experience ALTERNATE APPROACH:
a gravitational force of
Gm1m2
135 N. Determine the
Fg =
r2
new force, given the
following diagrams.
Change made Effect on F
g
BEFORE
10 kg
30 kg
r
ORIGINAL
135 N
Double m1
270 N
r/5
AFTER
20 kg
6.75 x 103 N
30 kg
r/5
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
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The Cavendish Experiment
Cavendish used a torsion balance to find G:
G=
|Fg|r2
The Cavendish Experiment
Cavendish’s torsion balance featured:
1) fixed heavy masses (m1) on each side
m1m2
The unit for G is N—m2/kg2 from this equation
2) hanging light masses (m2) connected to
the ends of a torsional pendulum
He measured Fg between two known masses
(m1 and m2) separated by distance r
3) torsion force (Fg) measured from the
movement of the hanging masses
The constant, G, could then be calculated
using Fg, m1, m2 and r
4) distance (r) between the centers of the
lead mass and the brass mass
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
The Cavendish Experiment
The Cavendish Experiment
Cavendish’s torsion balance:
Other versions:
TOP VIEW
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
Dulku – Physics 20 – Unit 2 (Dynamics) – Topic M
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