Final answer:
To calculate the normal component of acceleration for a particle on curve R(t), we differentiate R(t) to get the velocity and acceleration vectors, and then use the cross product at the specified time. Without the exact velocity given, we cannot compute the normal component directly from the provided acceleration at t = 8.
Step-by-step explanation:
To find the normal component of acceleration of a particle moving along a given curve, we need to calculate the acceleration vector and its component perpendicular to the velocity vector at the specified time. For the curve R(t) = cos(2t)i + sin(2t)j + 8t^2k, we first differentiate to obtain the velocity vector V(t) and then differentiate once more to get the acceleration vector A(t).
However, the question includes the acceleration vector at t = 8 seconds, which we'll use to find the normal component at that instant:
- A(t) = -4sin(2t)i + 4cos(2t)j+ 128tk
- A(8) = -4sin(16)i + 4cos(16)j + 1024k
To compute the normal component, we would cross multiply the acceleration and velocity vectors to get the direction perpendicular to both, and then take the magnitude of that product.
The normal acceleration is the component of the full acceleration that is perpendicular to the velocity vector, so it does not change the speed but only the direction of the particle. In this case, we need to calculate the velocity and acceleration vectors at t = 8, and then use the formula for normal acceleration an = |A(t) x V(t)| / |V(t)|. Unfortunately, without the exact velocity vector at t = 8, we cannot compute the exact value for the normal component, as it was not provided in the initial data.