Question

A dense particle with mass 7 kg follows the path r(t)= (sin(8t), cos(5t), 2t7/2) with units in meters and seconds. What force acts on the mass at t = 0?

0.00 kg m/s²

Answer 1

To find the force acting on the mass at t = 0, we calculate the second derivative of the position vector to get acceleration and then use Newton's second law. The resulting force at t = 0 is (0, -175, 0) Newtons.

Step-by-step explanation:

To determine the force acting on the mass at t = 0, we must evaluate the second derivative of the position vector r(t), which gives the acceleration, and then apply Newton's second law, F = ma. The given position function is r(t) = (sin(8t), cos(5t), 2t7/2) meters.

First, find the velocity by differentiating each component of r(t) with respect to time t:

• v(t) = r'(t) = (8cos(8t), -5sin(5t), 7t6/2) m/s

Then, find the acceleration vector by differentiating

v(t)

with respect to time t:

• a(t) = v'(t) = (-64sin(8t), -25cos(5t), 7*6/2*t5/2) m/s2

Evaluating a(t) at t = 0 gives:

• a(0) = (-64sin(0), -25cos(0), 0) = (0, -25, 0) m/s2

Finally, apply Newton's second law to find the force at t = 0:

• F = ma = 7 kg * (0, -25, 0) = (0, -175, 0) N

Therefore, the force acting on the mass at t = 0 is (0, -175, 0) Newtons.