A car travels in a flat circle of radius R. At a certain instant, the velocity of the car is 24 m/s west, and the total acceleration of the car is 2.5
m/s2, 53° north of west.

a) Find the radial and tangential components of the acceleration of the car at that moment.

b) If the car's tangential acceleration is constant, how long will it take for it to make one full circle from the point at which its velocity is 24m/s west?

a.

Radial acceleration = 1.997 m/s^2

Tangential acceleration = 1.505 m/s^2

b.

It will take 35.6 sec to make one full circle.

Explanation:

Given:

A car travels in a flat circle of radius R.

Velocity of the car, v = 24 m/s west.

Total acceleration, a = 2.5 m/s^2 NW

a.

We have to find the radial and tangential components of acceleration of the car at that moment.

From the figure shown we can say that:

⇒                        ⇒

⇒               ⇒

⇒                 ⇒

b.

Now we have to find the time taken for the angular displacement .

We know that:

⇒ and    so,

Let  t  be the time taken for the angular displacement  θ  to be  2 π .

 2 π  as it has taken full circle.

So,

⇒         ...equation (i)

⇒ , , ...

⇒

⇒

⇒ Plugging all in equation (i)

⇒

⇒

⇒

⇒

⇒ Solving the above quadratic.

⇒  where , and

⇒

⇒ and

⇒ Discarding the negative values.

⇒ time taken = 35.6 sec

Time taken by the car to make one full circle is 35.6 seconds.