A body of mass m moves under the influence of force F(r). Given it's trayectory r(t), find F(r).
F(t) = M * d^r/dt^2
If you want it as a function of r, use the inverse of the r(t) relation.
What do you mean with the inverse?
To find the force F(r) experienced by a body of mass m moving along a trajectory r(t), we need to differentiate the trajectory with respect to time twice. By doing this, we can obtain the acceleration of the body, which is proportional to the force acting on it according to Newton's second law of motion (F = ma).
Here are the steps to get the force F(r):
1. Start by differentiating the trajectory r(t) twice with respect to time t to obtain the acceleration a(t):
a(t) = d^2r(t)/dt^2
The first derivative of r(t) with respect to time gives the velocity v(t) of the body.
2. Next, multiply the acceleration a(t) by the mass m to get the force F(t):
F(t) = m * a(t)
Now we have the force F(t) as a function of time.
3. Finally, replace the variable t with the position vector r in terms of position rather than time to obtain the force F(r):
F(r) = m * a(r)
This gives us the force F(r) experienced by the body at any position along its trajectory.
Note that to fully determine F(r), you need additional information such as the form of the trajectory r(t) or any other specific conditions mentioned in the problem.