Answer

Verified

372.9k+ views

**Hint:**Here, using the angular velocity formula, which is the angle per unit time, the angle through which the rigid body rotates before it comes to rest is computed. The angular velocity is zero at rest, so we will calculate the time and put it in the angle equation.

**Complete Step-by-Step Solution:**

Let us suppose the angle through which it rotates be \[\theta \]

According to the question,

The angular velocity is given as\[\omega = a - bt\]

Now, if the body is at rest, the angular velocity of such a body is zero, i.e., \[\omega = 0\]

We can now equate the value of angular velocity such that

\[a - bt = 0\]

Upon solving, we get

\[t = \dfrac{a}{b}\]

Also, we know that angular velocity is also measured as angle per unit time, therefore,

$\omega = \dfrac{{d\theta }}{{dt}}$

Rearranging the terms, we get

\[d\theta = \omega dt\]

Now, we will integrate both the sides

\[\int\limits_0^\theta {d\theta } = \int\limits_0^{\dfrac{a}{b}} {\omega dt} \]

Upon further solving, we get

\[\theta = \int\limits_0^{\dfrac{a}{b}} {(a - bt) dt} \]

Now, let us integrate the above expression

\[\theta = \left[ {at - \dfrac{{b{t^2}}}{2}} \right]_0^{\dfrac{a}{b}}\]

\[ \Rightarrow \theta = \dfrac{{{a^2}}}{b} - \dfrac{{{a^2}}}{{2b}}\]

Let us simplify and get the final answer

\[\therefore \theta = \dfrac{{{a^2}}}{{2b}}\]

**Hence, the correct option is (B.)**

**Additional Information:**Angular velocity is the quantity of a vector and is described as the angular displacement change rate that specifies the object's angular velocity or rotational velocity and the axis around which the object is rotating. At a given period of time, the amount of change in the angular displacement of the particle is called angular velocity. The angular velocity vector's track is vertical to the rotation plane, in a direction usually indicated by the right-hand rule.

**Note:**In the above question, we have found out the angle by integrating the equation of \[\theta \] between the limits \[0\] to \[\dfrac{a}{b}\]. We could have also landed to the same result by putting the value of $t$ in the $\theta $ equation after integrating it.

Recently Updated Pages

Draw a labelled diagram of DC motor class 10 physics CBSE

A rod flies with constant velocity past a mark which class 10 physics CBSE

Why are spaceships provided with heat shields class 10 physics CBSE

What is reflection Write the laws of reflection class 10 physics CBSE

What is the magnetic energy density in terms of standard class 10 physics CBSE

Write any two differences between a binocular and a class 10 physics CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Name 10 Living and Non living things class 9 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail