Final answer:
The force acting on a mass of 7 kg at t = 0, given its position function r(t), is calculated to be (0, -567, 0) newtons.
Step-by-step explanation:
To find the force acting on a mass at a specific time given its position as a function of time, we need to differentiate the position function to find the velocity and then differentiate the velocity to find the acceleration. Applying Newton's second law, Force (F) is equal to mass (m) multiplied by acceleration (a). Since the student's question specifies the mass as 7 kg and provides the position function r(t) = (sin(3t), cos(9t), 2t9/2), we'll first differentiate this function to find the velocity vector v(t), then differentiate again to find the acceleration vector a(t) at t = 0.
The velocity vector v(t) is the derivative of r(t) with respect to time t:
v(t) = r'(t) = (3cos(3t), -9sin(9t), 9t8/2)
Next, the acceleration vector a(t) is the derivative of v(t) with respect to time t:
a(t) = v'(t) = (-9sin(3t), -81cos(9t), 36t7/2)
At t = 0, the acceleration vector a(0) becomes:
a(0) = (-9sin(0), -81cos(0), 0) = (0, -81, 0)
Finally, applying Newton's second law, the force vector F at t = 0 is:
F = ma = 7 kg * (0, -81, 0) = (0, -567, 0) N
The force acting on the mass at t = 0 is (0, -567, 0) newtons.