357
13-2 GRAVITATION AND THE PRINCIPLE OF SUPERPOSITION
13-2 GRAVITATION AND THE PRINCIPLE OF SUPERPOSITION
After reading this module, you should be able to . . .
13.04 If more than one gravitational force acts on a particle,
draw a free-body diagram showing those forces, with the
tails of the force vectors anchored on the particle.
13.05 If more than one gravitational force acts on a particle,
find the net force by adding the individual forces as
vectors.
● Gravitational forces obey the principle of superposition;
that is, if n particles interact, the net force
on a particle
labeled particle 1 is the sum of the forces on it from all the
other particles taken one at a time:
in which the sum is a vector sum of the forces
on
particle 1 from particles 2, 3, . . . , n.
F
:
1i
F
:
1,net ϭ ͚
n
iϭ2
F
:
1i ,
F
:
1,net
● The gravitational force
on a particle from an extended
body is found by first dividing the body into units of differential
mass dm, each of which produces a differential force
on
the particle, and then integrating over all those units to find
the sum of those forces:
F
:
1 ϭ ͵dF
:
.
dF
:
F
:
1
Learning Objectives
Key Ideas
Gravitation and the Principle of Superposition
Given a group of particles, we find the net (or resultant) gravitational force on
any one of them from the others by using the principle of superposition. This is a
general principle that says a net effect is the sum of the individual effects. Here,
the principle means that we first compute the individual gravitational forces that
act on our selected particle due to each of the other particles. We then find the net
force by adding these forces vectorially, just as we have done when adding forces
in earlier chapters.
Let’s look at two important points in that last (probably quickly read) sentence. (1) Forces are vectors and can be in different directions, and thus we must
add them as vectors, taking into account their directions. (If two people pull on
you in the opposite direction, their net force on you is clearly different than if
they pull in the same direction.) (2) We add the individual forces. Think how impossible the world would be if the net force depended on some multiplying factor
that varied from force to force depending on the situation, or if the presence of
one force somehow amplified the magnitude of another force. No, thankfully, the
world requires only simple vector addition of the forces.
For n interacting particles, we can write the principle of superposition for the
gravitational forces on particle 1 as
1,net ϭ 12 ϩ 13 ϩ 14 ϩ 15 ϩ и и и ϩ 1n .
( 1 3 - 4 )
Here 1,net is the net force on particle 1 due to the other particles and, for example, 13 is the force on particle 1 from particle 3. We can express this equation
more compactly as a vector sum:
(13-5)
Real Objects. What about the gravitational force on a particle from a real (extended) object? This force is found by dividing the object into parts small enough to
treat as particles and then using Eq. 13-5 to find the vector sum of the forces on the
particle from all the parts. In the limiting case, we can divide the extended object
into differential parts each of mass dm and each producing a differential force dF
:
F
:
1,net ϭ ͚
n
iϭ2
F
:
1i .
F
:
F
:
F
:
F
:
F
:
F
:
F
:
F
:
13-2 GRAVITATION AND THE PRINCIPLE OF SUPERPOSITION
13-2 GRAVITATION AND THE PRINCIPLE OF SUPERPOSITION
After reading this module, you should be able to . . .
13.04 If more than one gravitational force acts on a particle,
draw a free-body diagram showing those forces, with the
tails of the force vectors anchored on the particle.
13.05 If more than one gravitational force acts on a particle,
find the net force by adding the individual forces as
vectors.
● Gravitational forces obey the principle of superposition;
that is, if n particles interact, the net force
on a particle
labeled particle 1 is the sum of the forces on it from all the
other particles taken one at a time:
in which the sum is a vector sum of the forces
on
particle 1 from particles 2, 3, . . . , n.
F
:
1i
F
:
1,net ϭ ͚
n
iϭ2
F
:
1i ,
F
:
1,net
● The gravitational force
on a particle from an extended
body is found by first dividing the body into units of differential
mass dm, each of which produces a differential force
on
the particle, and then integrating over all those units to find
the sum of those forces:
F
:
1 ϭ ͵dF
:
.
dF
:
F
:
1
Learning Objectives
Key Ideas
Gravitation and the Principle of Superposition
Given a group of particles, we find the net (or resultant) gravitational force on
any one of them from the others by using the principle of superposition. This is a
general principle that says a net effect is the sum of the individual effects. Here,
the principle means that we first compute the individual gravitational forces that
act on our selected particle due to each of the other particles. We then find the net
force by adding these forces vectorially, just as we have done when adding forces
in earlier chapters.
Let’s look at two important points in that last (probably quickly read) sentence. (1) Forces are vectors and can be in different directions, and thus we must
add them as vectors, taking into account their directions. (If two people pull on
you in the opposite direction, their net force on you is clearly different than if
they pull in the same direction.) (2) We add the individual forces. Think how impossible the world would be if the net force depended on some multiplying factor
that varied from force to force depending on the situation, or if the presence of
one force somehow amplified the magnitude of another force. No, thankfully, the
world requires only simple vector addition of the forces.
For n interacting particles, we can write the principle of superposition for the
gravitational forces on particle 1 as
1,net ϭ 12 ϩ 13 ϩ 14 ϩ 15 ϩ и и и ϩ 1n .
( 1 3 - 4 )
Here 1,net is the net force on particle 1 due to the other particles and, for example, 13 is the force on particle 1 from particle 3. We can express this equation
more compactly as a vector sum:
(13-5)
Real Objects. What about the gravitational force on a particle from a real (extended) object? This force is found by dividing the object into parts small enough to
treat as particles and then using Eq. 13-5 to find the vector sum of the forces on the
particle from all the parts. In the limiting case, we can divide the extended object
into differential parts each of mass dm and each producing a differential force dF
:
F
:
1,net ϭ ͚
n
iϭ2
F
:
1i .
F
:
F
:
F
:
F
:
F
:
F
:
F
:
F
: