Rotational Motion

Motion of Center of Mass

The center of mass of a system of particles moves as if all the mass of the system was concentrated at the center of mass and all the external forces were applied at that point.

If M is the total mass of the body and V is its velocity then,

`MV=m_1v_1+m_2v_2+m_3v_3+..`

If A is acceleration, then

`MA=m_1a_1+m_2a_2+m_3a_3+…`

We know that Force = mass × acceleration. So,

`MA=F_1+F_2+F_3+….F_n`

Let us consider the force F₁ being applied on particle P₁. It is not a single force rather sum of all the forces on the given particle. Each particle will experience external as well as internal forces. According to Newton’s Third Law of motion, internal forces will be equal and opposite to external forces. So, contribution of internal forces will be zero. Only the external forces contribute to the equation. So, previous equation can be written as follows:

MA = F_ext

Here, F_ext is sum of all the external forces acting on particles of the system.

Linear Momentum of a System of Particles

Linear momentum `p=mv`

For a system of n number of particles:

`P=p_1+p_2+p_3+…p_n`

`=m_1v_1+m_2v_2+m_3v_3+…m_n\v_n`

Or, `P=MV`

So, total momentum of a system of particles is equal to the product of total mass and velocity of the center of mass.

Vector Product of Two Vectors

A vector product of two vectors a and b is a vector c such that

Magnitude of c = c = ab sin θ where a and b are magnitudes of a and b and θ is the angle between two vectors.

c is perpendicular to the plane containing a and b.

Right Hand Rule

Open up your right hand palm and curl your fingers pointing from a to b. If direction of fingers point to direction of rotation of one vector with respect to another, then direction of thumb shows the direction of the vector product.

Vector product is not commutative, i.e. a × b≠ b × a.

a × b = - b × a

This means that when we change the direction of rotation, the vector products are in opposite directions.

Under reflection, all the components of a vector change sign so, under reflection:

a × b → (-a) × (-b) = a × b

Both scalar and vector products show distributive property of addition.

a.(b + c) = a.b + a.c

a × a = 0 because a² sin 0° = 0

Angular Velocity and Its Relation with Linear Velocity

Angular velocity of a particle is given by following equation:

`v=ωr`

Where, ω (Greek letter omega) is angular velocity, v is linear velocity and r is the radius of circular path on which the particle is moving.

Let us assume that a particle is at a perpendicular distance r_i from the fixed axis and the linear velocity at a given instant is v_i. Then

`v_i=ωr_i`

This equation shows that angular velocity is same for all the particles. So, ω is referred to as the angular velocity of the whole body. Angular velocity vector lies along the axis of rotation and points out in the direction in which the right handed screw would advance when head of screw is rotated with the body.

This figure shows a particle P on a circular path around a fixed axis (z-axis). Vector ω is directed along the z-axis. Position vector r = OP

So, `ω×r=ω×OP`

`=ω×(OC+CP)`

Since ω is along OC so ω × OC = 0

So, `ω×r=ω×CP`

The vector ω × CP is perpendicular to ω and also to CP. So, it is along the tangent to the circle at P.

Angular acceleration

Angular acceleration α is the time rate of change of angular velocity.

`α=(dω)/(dt)`