48
2 Particle Dynamics
are found to be valid in that frame. It is found that inertial frames move with constant
velocity with respect to one another.
Conservation Laws
(i) If the total force F is zero, then linear momentum p is conserved.
(ii) If the total external torque is zero, then the angular momentum J is conserved.
(iii) If the forces acting on a particle are conservative, then the total mechanical
energy (kinetic + potential) of the particle is conserved.
Conservative Force
If the force field is such that the work done around a closed orbit is zero, i.e.
F · ds = 0
(2.1)
then the force and the system are said to be conservative. A system cannot be conservative if a dissipative force like friction is present. Since the quantity Fds due to
friction will always be negative and the integrand cannot vanish, by Stokes theorem
the condition for conservative forces given by (2.1) becomes
∇ × F = 0
(2.2)
Since the curl of a gradient always vanishes, it follows that F must be the gradient
of the scalar quantity V , i.e.
F = −∇V
(2.3)
V is called the potential energy.
Centre of Mass
r c =
1
M
n
i=1
m i r i
(2.4)
where r c is the position vector of the centre of mass from the origin and M = m i
is the total mass. The centre of mass moves as if it were a single particle of mass
equal to the total mass of the system, acted upon by the total external force and
independent of the nature of the internal forces.
The reduced mass (μ) of two bodies of mass m 1 and m 2 is given by
μ =
m 1 m 2
m 1 + m 2
(2.5)
2 Particle Dynamics
are found to be valid in that frame. It is found that inertial frames move with constant
velocity with respect to one another.
Conservation Laws
(i) If the total force F is zero, then linear momentum p is conserved.
(ii) If the total external torque is zero, then the angular momentum J is conserved.
(iii) If the forces acting on a particle are conservative, then the total mechanical
energy (kinetic + potential) of the particle is conserved.
Conservative Force
If the force field is such that the work done around a closed orbit is zero, i.e.
F · ds = 0
(2.1)
then the force and the system are said to be conservative. A system cannot be conservative if a dissipative force like friction is present. Since the quantity Fds due to
friction will always be negative and the integrand cannot vanish, by Stokes theorem
the condition for conservative forces given by (2.1) becomes
∇ × F = 0
(2.2)
Since the curl of a gradient always vanishes, it follows that F must be the gradient
of the scalar quantity V , i.e.
F = −∇V
(2.3)
V is called the potential energy.
Centre of Mass
r c =
1
M
n
i=1
m i r i
(2.4)
where r c is the position vector of the centre of mass from the origin and M = m i
is the total mass. The centre of mass moves as if it were a single particle of mass
equal to the total mass of the system, acted upon by the total external force and
independent of the nature of the internal forces.
The reduced mass (μ) of two bodies of mass m 1 and m 2 is given by
μ =
m 1 m 2
m 1 + m 2
(2.5)