The wheel rotates with a constant angular acceleration = 3 rad / s2. Determine the radius of the wheel

The wheel rotates with a constant angular acceleration = 3 rad / s2. Determine the radius of the wheel if its linear acceleration through t = 1 s from the start of movement is 7.5 m / s2.

Given:
Angular acceleration e = 3 rad / s ^ 2;
Time t = 1 s;
Speed v = 7.5 m / s ^ 2;
Find the radius of the wheel r.
Decision:
Apply the formula a = √ (a (r) ^ 2 + a (n) ^ 2);
To calculate a (r) and a (n), we apply the formulas:
a (r) = e * r;
a (n) = v ^ 2 / r = w ^ 2 * r = w ^ 2 * t ^ 2 * r;
Substitute the known values into the formula and calculate the radius.
a = √ (a (r) ^ 2 + a (n) ^ 2) = √ ((e * r) ^ 2 + (v ^ 2 / r) ^ 2);
Let’s square it.
a ^ 2 = (e * r) ^ 2 + (w ^ 2 * t ^ 2 * r) ^ 2;
a ^ 2 = e ^ 2 * r ^ 2 + w ^ 4 * t ^ 4 * r ^ 2;
Hence:
r = a / (e * √ (1 + e ^ 2 * t ^ 4);
We get:
r = 10 / (3 * √ (1 + 9 * 1) = 79 cm.