A horizontal disk with a radius of 23 m ro- tates about a vertical axis through its center. The disk starts from rest and has a constant angular acceleration of 5.5 rad/s2.

At what time will the radial and tangen- tial components of the linear acceleration of a point on the rim of the disk be equal in magnitude?

To find the time at which the radial and tangential components of the linear acceleration on the rim of the disk are equal in magnitude, we need to understand the relationship between angular acceleration and linear acceleration.

In circular motion, the linear acceleration of a point on the rim of a rotating disk can be broken down into two components: radial acceleration (ar) and tangential acceleration (at).

Radial acceleration (ar) is directed towards the axis of rotation and is given by the formula:

ar = r * α

where r is the radius of the disk and α is the angular acceleration.

Tangential acceleration (at) is perpendicular to the radial direction and is given by the formula:

at = r * ω^2

where ω is the angular velocity of the disk.

In this case, the angular acceleration α is given as 5.5 rad/s^2, and we know that the disk starts from rest. Therefore, the initial angular velocity ωi is 0.

To find the time at which the radial and tangential components of the linear acceleration are equal in magnitude, we need to equate the two expressions for acceleration, ar and at:

r * α = r * ω^2

Canceling out the radius r from both sides of the equation, we get:

α = ω^2

Now we can solve for the angular velocity ω. Rearranging the equation, we have:

ω = sqrt(α)

Substituting the given value of angular acceleration α = 5.5 rad/s^2, we calculate:

ω = sqrt(5.5)

Finally, we can find the time at which the radial and tangential components of linear acceleration are equal by dividing the angular acceleration by the angular velocity:

t = α / ω

Substituting the values, we get:

t = 5.5 / sqrt(5.5)

Calculating this, we find the time at which the radial and tangential components of the linear acceleration on the rim of the disk are equal in magnitude.