IIT JEE , AIEEE Forum - Physics , Mechanics

antonyajay21

Why is average and instantaneous angular accelerations, vectors?Whereas average angular velocity is a scalar and instantaneous angular velocity is a vector...

anandghegde

ROTATIONAL QUANTITIES AS VECTORS:
Angular motion has direction associated with it and is inherently a vector process. But a point on a rotating wheel is continuously changing its direction and it is inconvenient to track that direction. The only fixed, unique direction for a rotating wheel is the axis of rotation so, it is logical to choose axial direction as the direction of the angular velocity.Left with two choices about direction, it is customary to use the right hand rule to specify the direction of angular quantities, as shown in figure

anandghegde

in fig 1 Right hand rule: turn the fingers
of your right hand along the sense of rotation,
the direction in which the thumb points is the
direction of the angular velocity. In this case
the direction of the angular velocity of the disc
is pointing perpendicularly upwards from the
plane of rotation of the disc

in fig 2 In this case the direction of the
angular velocity of the disc is pointing
perpendicularly downwards from the plane
of rotation of the disc.

anandghegde

Here you should note that nothing moves in the direction of angular velocity, i.e., nothing moves along the axis of rotation. The direction of angular velocity represents the rotational motion taking place in the plane perpendicular to
the axis. In a similar way we can associate a direction with angular acceleration also. If the angular velocity is increasing then the direction of the angular acceleration coincides with that of the angular velocity as shown in figure
3 and if the angular velocity is decreasing then the direction of the angular acceleration is opposite to that of angular velocity, as shown in figure 4

anandghegde

Now, we can say that the angular velocity and the angular acceleration are vector quantities. As their directions coincide with the axis of rotation, these quantities are also called axial vectors.

Angular velocity and angular acceleration follow the triangle law of vector addition and their additions are commutative also but the addition of angular displacements are noncommutative, hence angular displacement is not a vector quantity. Noncommutative nature of the angular displacement is shown in figure

anandghegde

The orientation in the sixth figure is not the same as in the third. Obviously the commutative law of addition is not satisfied by these rotations. Despite the fact that they have a magnitude and a direction, finite rotations cannot be represented as vectors

anandghegde

But, as the two angular displacements are made smaller, the difference between two sums disappear rapidly. If the angular displacements are made infinitesimal, the order of addition no longer affects the result. Hence infinitesimal
angular displacements are vectors.

antonyajay21

Thanx dude,it was worth reading.But my doubt isn't cleared still.Why is instantaneous acceleration and average acceleration vectors???Whereas instantaneous velocity is a vector and average velocity is not.

anchitsaini

pls tell me where did u read that
because uptil now i thought average velocity was a vector!!!

antonyajay21

Average angular velocity is not a vector.Check it out in Modern ABC of class 11th by Satish .K.Gupta

anchitsaini

yes u are correct
but the explanation given is also sufficient
since average ang vel=delta theta/delta t
delta theta=delta t*ang vel
scalar=scalar*?
if ? is vector than lhs would be a vector which is not true hence its a scalar
similarly--
average ang acc=delta w/delta t
delta w=delta t *av ang acc
vector=scalar *?
if ? is a scalar lhs would be a scalar which is not true hence its a vector

antonyajay21

Thanx guys!!!!!

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