Final answer:
To find the normal component of acceleration at t=8 for a particle moving in a curve given by r(t)=cos3ti+sin3tj+4tk, we need to find the acceleration vector and calculate its projection onto the vector perpendicular to the velocity vector.
Step-by-step explanation:
The normal component of acceleration is the component of acceleration perpendicular to the velocity vector. In this case, we have the position vector r(t) = cos(3t)i + sin(3t)j + 4tk. To find the normal component of acceleration at t = 8, we need to find the acceleration vector and then calculate its projection onto the vector perpendicular to the velocity vector.
First, we find the velocity vector v(t) by taking the derivative of r(t) with respect to t. v(t) = -3sin(3t)i + 3cos(3t)j + 4k.
Next, we find the acceleration vector a(t) by taking the derivative of v(t) with respect to t. a(t) = -9cos(3t)i - 9sin(3t)j.
Finally, we calculate the projection of a(t) onto the vector perpendicular to v(t). The normal component of acceleration at t = 8 is the magnitude of the projection vector.