Final answer:
The force required so that a particle of mass m has the position function r(t) = t³ i + 8t² j + t³ k is obtained by first finding the particle's acceleration as the second derivative of the position function and then applying Newton's second law, F=ma, to get F(t) = (6mt i + 16m j + 6mt k) N.
Step-by-step explanation:
The question asks what force is required so that a particle of mass m has the position function r(t) = t³ i + 8t² j + t³ k. To find this force, we need to first determine the acceleration of the particle, which is the second derivative of the position function with respect to time. Mathematically this is represented as a(t) = d²r/dt². From the given position function, we have:
- r(t) = t³ i + 8t² j + t³ k
- The velocity function is v(t) = dr/dt = 3t² i + 16t j + 3t² k
- The acceleration function is a(t) = d²r/dt² = 6t i + 16 j + 6t k
By Newton's second law of motion, F = ma, where F is the force, m is the mass, and a is the acceleration. Hence, the required force can be found by multiplying the acceleration function by the mass of the particle:
F(t) = m × (6t i + 16 j + 6t k)
We then multiply each component of the acceleration by the mass to obtain the components of the force:
F(t) = (6mt i + 16m j + 6mt k) N