A body of mass m moves under the influence of force F(r). Given it's trayectory r(t), find F(r).
To find the force F(r) acting on a body of mass m, you need to differentiate the trajectory r(t) twice with respect to time (t). The force acting on the body will be proportional to the second derivative of the trajectory:
F(r) = m * d²r(t)/dt²
Here's a step-by-step explanation of how to find F(r) using the given trajectory r(t):
1. Start by differentiating the trajectory r(t) once with respect to time to find the velocity v(t). This can be represented as:
v(t) = dr(t)/dt
2. Next, differentiate the velocity v(t) obtained in step 1 with respect to time to find the acceleration a(t). This can be represented as:
a(t) = dv(t)/dt = d²r(t)/dt²
3. Finally, substitute the acceleration a(t) obtained in step 2 and the mass m into the equation F(r) = m * d²r(t)/dt² to obtain the force F(r) acting on the body:
F(r) = m * a(t)
By following these steps and calculating the acceleration using the given trajectory, you can find the force F(r) acting on the body.