Note that this result is derived from a vector equation, p = 0, and therefore applies for each component of the linear momentum. To state the result in other terms, we let s be some constant vector such that F ·s = 0, independent of time. Then p's = F.s = 0 or, integrating with respect to time, p's = constant (2.80) which states that the component of linear momentum in a direction in which the force vanishes is constant in time. The angular momentum L of a particle with respect to an origin from which the position vectorris measured is defined to be L=rx p (2.81) The torque or moment of force N with respect to the same origin is defined to be N =r x F (2.82) where r is the position vector from the origin to the point where the force F is applied. Because F = mv for the particle, the torque becomes N = r x mỷ = r x Þ Now L = (r x x p) = († x p) + (r × p) but i xp = i x mv = m(† × i) = 0 so L = r x p = N (2.83) If no torques act on a particle (i.e., if N = 0), then L = 0 and L is a vector con- stant in time. The second important conservation theorem is
Note that this result is derived from a vector equation, p = 0, and therefore applies for each component of the linear momentum. To state the result in other terms, we let s be some constant vector such that F ·s = 0, independent of time. Then p's = F.s = 0 or, integrating with respect to time, p's = constant (2.80) which states that the component of linear momentum in a direction in which the force vanishes is constant in time. The angular momentum L of a particle with respect to an origin from which the position vectorris measured is defined to be L=rx p (2.81) The torque or moment of force N with respect to the same origin is defined to be N =r x F (2.82) where r is the position vector from the origin to the point where the force F is applied. Because F = mv for the particle, the torque becomes N = r x mỷ = r x Þ Now L = (r x x p) = († x p) + (r × p) but i xp = i x mv = m(† × i) = 0 so L = r x p = N (2.83) If no torques act on a particle (i.e., if N = 0), then L = 0 and L is a vector con- stant in time. The second important conservation theorem is
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Proved the 2.83 equations
Transcribed Image Text:Note that this result is derived from a vector equation, p = 0, and therefore
applies for each component of the linear momentum. To state the result in
other terms, we let s be some constant vector such that F ·s = 0, independent of
time. Then
p's = F.s = 0
or, integrating with respect to time,
p's = constant
(2.80)
which states that the component of linear momentum in a direction in which the force
vanishes is constant in time.
The angular momentum L of a particle with respect to an origin from which
the position vectorris measured is defined to be
L=rx p
(2.81)
The torque or moment of force N with respect to the same origin is defined
to be
N =r x F
(2.82)
where r is the position vector from the origin to the point where the force F is
applied. Because F = mv for the particle, the torque becomes
N = r x mỷ = r x Þ
Now
L = (r x
x p) = († x p) + (r × p)
but
i xp = i x mv = m(† × i) = 0
so
L = r x p = N
(2.83)
If no torques act on a particle (i.e., if N = 0), then L = 0 and L is a vector con-
stant in time. The second important conservation theorem is
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