Angular position of the particle
If you visualize a particle moving in a radius of r with O as the centre of the circle. That means OP is equal to r. With O as origin mark OX on the X axis.
The position of P can be described by angle θ between Op and OX.
Angle θ is called the angular position of the particle
The rate of change of angular position is called angular velocity.
ω = dθ/dt
The rate of change of angular velocity is angular acceleration.
Radial unit vector
Tangential unit vector
7.5 Circular turnings and banking of roads
When vehicles go through turnings, they travel along a nearly circular arc. That means there is centrepetal accelaration. What forces cause this acceleration? Friction fs can act towards the centre. However this may not be sufficient and the vehicle may skid.
To take care of it, the roads are banked t the turn so that outer part of the road is somewhat lifted up as compared to the inner part. therefore the normal force makes an angle θ wit hte vertical. The horizontal component of the normal force helps in providing the accelaration required.
The θ required for a speed of the vehicle of v is given by
tanθ = v²/rg
7.6 Centrifugal force
What psuedo force is required if the frame of reference rotates at a constant angular velocity ω with respect to an inertial frame?
Note: It is a common minsconception among the beginners that centrifugal force acts on a particle because the particle goes on a circle. Centrigual force acts (or is assumed to act) becasue we describe the particle from a rotating frame which is noninertial and still use Newton's laws.
7.7 Effect of earth's rotation on apprarent weight
A plumb line stays in a direction which is different from true vertical to earth at that point. The walls of building are built by making them parallel to the plumb line and not to the true vertical.
The weight of a body is mg' and not mg and g' is less than g.
g' = g only at the poles as the poles themselved do not rotate and hence the effect of earth's rotation is not felt there.