307
11-6 NEWTON’S SECOND LAW IN ANGULAR FORM
11-6 NEWTON’S SECOND LAW IN ANGULAR FORM
After reading this module, you should be able to . . .
11.21 Apply Newton’s second law in angular form to relate the torque acting on a particle to the resulting rate of change of the
particle’s angular momentum, all relative to a specified point.
● Newton’s second law for a particle can be written in angular form as
where
is the net torque acting on the particle and is the angular momentum of the particle.
ᐉ
:
t
:
net
t
:
net ϭ
dᐉ
:
dt
,
Learning Objective
Key Idea
Newton’s Second Law in Angular Form
Newton’s second law written in the form
(single particle)
(11-22)
expresses the close relation between force and linear momentum for a single
particle. We have seen enough of the parallelism between linear and angular
quantities to be pretty sure that there is also a close relation between torque
and angular momentum. Guided by Eq. 11-22, we can even guess that it must be
(single particle).
(11-23)
Equation 11-23 is indeed an angular form of Newton’s second law for a single particle:
t
:
net ϭ
dᐉ
:
dt
F
:
net ϭ
dp
:
dt
The (vector) sum of all the torques acting on a particle is equal to the time rate of
change of the angular momentum of that particle.
Equation 11-23 has no meaning unless the torques and the angular momentum
are defined with respect to the same point, usually the origin of the coordinate
system being used.
Proof of Equation 11-23
We start with Eq. 11-18, the definition of the angular momentum of a particle:
where is the position vector of the particle and is the velocity of the particle.
Differentiating* each side with respect to time t yields
(11-24)
However,
is the acceleration of the particle, and
is its velocity .
Thus, we can rewrite Eq. 11-24 as
dᐉ
:
dt
ϭ m(r
: ϫ a
: ϩ v
: ϫ v
:
).
v
:
dr
:
/dt
a
:
dv
:
/dt
dᐉ
:
dt
ϭ m r
: ϫ
dv
:
dt
ϩ
dr
:
dt
ϫ v
:
.
v
:
r
:
ᐉ
: ϭ m(r
: ϫ v
:
),
ᐉ
:
t
:
*In differentiating a vector product, be sure not to change the order of the two quantities (here and
) that form that product. (See Eq. 3-25.)
v
:
r
:
11-6 NEWTON’S SECOND LAW IN ANGULAR FORM
11-6 NEWTON’S SECOND LAW IN ANGULAR FORM
After reading this module, you should be able to . . .
11.21 Apply Newton’s second law in angular form to relate the torque acting on a particle to the resulting rate of change of the
particle’s angular momentum, all relative to a specified point.
● Newton’s second law for a particle can be written in angular form as
where
is the net torque acting on the particle and is the angular momentum of the particle.
ᐉ
:
t
:
net
t
:
net ϭ
dᐉ
:
dt
,
Learning Objective
Key Idea
Newton’s Second Law in Angular Form
Newton’s second law written in the form
(single particle)
(11-22)
expresses the close relation between force and linear momentum for a single
particle. We have seen enough of the parallelism between linear and angular
quantities to be pretty sure that there is also a close relation between torque
and angular momentum. Guided by Eq. 11-22, we can even guess that it must be
(single particle).
(11-23)
Equation 11-23 is indeed an angular form of Newton’s second law for a single particle:
t
:
net ϭ
dᐉ
:
dt
F
:
net ϭ
dp
:
dt
The (vector) sum of all the torques acting on a particle is equal to the time rate of
change of the angular momentum of that particle.
Equation 11-23 has no meaning unless the torques and the angular momentum
are defined with respect to the same point, usually the origin of the coordinate
system being used.
Proof of Equation 11-23
We start with Eq. 11-18, the definition of the angular momentum of a particle:
where is the position vector of the particle and is the velocity of the particle.
Differentiating* each side with respect to time t yields
(11-24)
However,
is the acceleration of the particle, and
is its velocity .
Thus, we can rewrite Eq. 11-24 as
dᐉ
:
dt
ϭ m(r
: ϫ a
: ϩ v
: ϫ v
:
).
v
:
dr
:
/dt
a
:
dv
:
/dt
dᐉ
:
dt
ϭ m r
: ϫ
dv
:
dt
ϩ
dr
:
dt
ϫ v
:
.
v
:
r
:
ᐉ
: ϭ m(r
: ϫ v
:
),
ᐉ
:
t
:
*In differentiating a vector product, be sure not to change the order of the two quantities (here and
) that form that product. (See Eq. 3-25.)
v
:
r
: