Calculate the maximum force value.

Question

The material point oscillation equation is x = 0.1sin (πt / 8 + π / 4) m; mass of this point m = 1.6 · 10 -2 kg. Display graphically the dependence between the force F acting on the point and the time t (within one period). Calculate the maximum force value. [24.6 · 10 -5 N].

Answer 1

Given the equation for x, we can calculate the acceleration of the point by taking the second derivative of x with respect to t. By taking the first derivative, we get the velocity, which is given by v = dx/dt = 0.1 * (pi/8) * cos(pi * t / 8 + pi/4). Taking the derivative of the velocity we get a = dv/dt = d^2x/dt^2 = -0.1 * (pi/8)^2 * sin(pi * t / 8 + pi/4). We know from Newton's second law that F=ma, so if we multiply the equation for a which we got by m, we get the force, so F = -1.6 * 10^-2 * (0.1) * (pi/8)^2 * sin(pi * t/8 + pi/4). We know that sin can take values from -1 to 1, so because F has a negative sign in front, it will be maximized for sin(pi*t/8 + pi/4) = -1. Plugging -1 in for sin(pi*t/8 + pi/4) and evaluating everything we get F_max = 2.467 * 10^-4.

Hope this helps! Let me know if I can clarify anything!