Question

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`?

Answer 1

`t = sqrt((2 pi)/9) "seconds"`

Explanation:

If you think of this as a linear problem, the magnitude of the velocity will simply be:
`|v| = |v_0| +|a*t|`
And the other equations of motion work in a similar way:
`d = v_0*t + 1/2 a*t^2`

The distance along the direction of travel is simply one eighth of a circle:
`d = 2 pi*r/8 = 2 pi * 4/8 = pi " meters"`

Replacing this value in the equation of motion for distance gives:
`pi = v_0*t + 1/2 a*t^2`
`pi = 0*t + 1/2 a*t^2`
`2 pi = a*t^2`
`2 pi = 9 * t^2`
`(2 pi)/9 = t^2`
`sqrt((2 pi)/9) = t`