Question

For circular motion, if $\overrightarrow {{a_t}}$, $\overrightarrow {{a_c}}$, $\overrightarrow r$ and $\overrightarrow v$ are tangential acceleration, centripetal acceleration, radius vector and velocity respectively, then find the wrong relation:
A. $\overrightarrow {{a_t}} .\overrightarrow {{a_c}} = 0$
B. $\overrightarrow {{a_t}} .\overrightarrow v {\text{ }}$ may be positive or negative
C. $\overrightarrow {{a_c}} .\overrightarrow v {\text{ }}$ may be positive or negative
D. $\overrightarrow {{a_c}} .\overrightarrow v = 0$

Answer 1

We need to define tangential acceleration, centripetal acceleration, radius vector and velocity before finding the relation between them. All the quantities are vector quantities and are associated with a direction along with its magnitude.

Complete answer:
We know that in a circular motion, the acceleration of the particle is said to be towards the centre and is perpendicular to the velocity vector. The acceleration is centripetal since the velocity is said to be tangential. Now if we look at the vector representation of a circular motion, the acceleration has two components namely radial and tangential.

The tangential acceleration is in the direction of the velocity vector and is perpendicular to the radial acceleration whereas the radial acceleration is along the radius of the circular motion. In a centripetal acceleration of the body, the acceleration which is directed along the radius of the path is called radial acceleration.

Now we can therefore figure out that the centripetal acceleration is perpendicular to the velocity vector as well as the tangential component of the acceleration. And we know that the dot product of two perpendicular vectors is zero therefore relation (A) and (D) are correct. Now the velocity vector can be in positive x-direction or negative x-direction and even the tangential acceleration can be in positive x-direction or negative x-direction therefore option (B) is also correct.

Hence we are left with only one incorrect relation. Thus, option C is the wrong relationship.

Note: In a circular motion the particle has acceleration and velocity as components of x and y axes. Velocity is concerning the x-axes and acceleration has two components namely tangential and radial. We should note that the dot product of two perpendicular vector quantities is zero.