- Moment of a Force (Motion of System of Particles and Rigid Body)
- Angular Momentum and Law of Conservation of Angular Momentum and Its Applications
- Moment of force (Torque)
- Angular momentum of a particle
- Torque and angular momentum for a system of particles
- conservation of angular momentum
Torque & Angular Momentum:
The rotational analogue of force is moment of force (Torque).
If a force acts on a single particle at a point P whose position with respect to the origin 'O' is given by the position vector r the moment of the force acting on the particle with respect to the origin 'O' is defined as the vector product t = r × F = rF sinΘ
Torque is vector quantity.
The moment of a force vanishes if either
The magnitude of the force is zero, or
The line of action of the force (r sinΘ) passes through the axis.
Example: Determine the torque on a bolt, if you are pulling with a force F directed perpendicular to a wrench of length 'l'cm?
Solution: t = r x F = rF sinΘ
In this case Θ=90o
The quantity angular momentum is the rotational analogue of linear momentum.
It could also be referred to as moment of (linear) momentum.
l = r × p
Rotational analogue of Newton’s second law for the translational motion of a single particle: `"dl"/"st" = tau`
Torque and angular momentum of system of particles:
The total angular momentum of a system of particles about a given point is addition of the angular momenta of individual particles added vectorially.
`"L" ="l"_1 + "l"_2 + ... + "l"_"n" = sum_(1)^nl_i ` where ` l_i=r_ixp_i`
Similarly for total torque on a system of particles is addition of the torque on an individual particle added vectorially.
`tau=sum_(1)^nr_i xx F_i = tau_"ext" + tau_"int"`
The torque resulting from internal forces is zero, due to
Newton’s third law i.e. these forces are equal and opposite.
These forces act on the line joining any two particles
The time rate of the total angular momentum of a system of particles about a point is equal to the sum of the external torques acting on the system taken about the same point.
`"dL"/"dt" = tau_"ext"`
Conservation of Angular Momentum
If the external torque acting on a system is zero, then its angular momentum remains conserved.
If text = 0, then dL/dt = 0 = I(ω) = constant ⇒ I1ω1= I2ω2