# Starting from rest, a particle is constrained to move in a circle of radius #4 m#. The tangential acceleration is #a_t = 9 m/s^2#. How long will it take to rotate #45º#?

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To find the time it takes for the particle to rotate 45º, we can use the equation for angular acceleration:

[ \alpha = \frac{{a_t}}{{r}} ]

Where:

- ( \alpha ) is the angular acceleration,
- ( a_t ) is the tangential acceleration, and
- ( r ) is the radius of the circle.

Given that ( a_t = 9 , \text{m/s}^2 ) and ( r = 4 , \text{m} ), we can calculate the angular acceleration:

[ \alpha = \frac{{9 , \text{m/s}^2}}{{4 , \text{m}}} ]

Now, we need to find the time it takes for the particle to rotate 45º. We know that the angular displacement ( \theta ) is 45º, which is equivalent to ( \frac{{\pi}}{{4}} ) radians. We can use the equation for angular displacement:

[ \theta = \frac{1}{2} \alpha t^2 ]

Where:

- ( \theta ) is the angular displacement,
- ( \alpha ) is the angular acceleration, and
- ( t ) is the time.

Rearranging the equation to solve for ( t ), we get:

[ t = \sqrt{\frac{{2 \theta}}{{\alpha}}} ]

Substituting ( \theta = \frac{{\pi}}{{4}} ) radians and the calculated value of ( \alpha ), we can find ( t ).

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