# Time of Motion with Negative Feedback

• SystemTheory
In summary, the conversation discusses the problem of computing the time it takes for a block of mass m to slide down an incline plane of length xL, with a constant friction factor b that causes the friction force to increase linearly with forward velocity v. Various approaches have been attempted, including solving for time in a differential equation and applying the Conservation of Energy principle, but a solution has not yet been found. The conversation also includes equations for differential work and power, but it is uncertain how these may contribute to solving the problem.
SystemTheory
A block of mass m slides down an incline plane of length xL. The friction factor b is constant, where the friction force increases as a linear function of forward velocity v.

This produces exponential motion. My problem in building an educational simulator is this, I want to compute the time down the ramp of length xL rather than make educated guesses in the simulator time setting.

I've tried solving for time in the DE, with no result. I've also tried to apply Conservation of Energy but I'm still struggling with an answer. Anyone recognize a solution to this apparently simple challenge?

I've posed the problem in this thread:

$$\frac{dv}{dt} = \frac{mg sin\theta - bv}{m}$$

$$\frac{d^2 x}{dt^2} = g\sin{\theta} - \frac{ b}{m} \frac{dx}{dt}$$

This is a nonhomogeneous linear 2nd order DE. The solution is (I think)
$$x(t)=A + Be^{-bt/m} + \frac{mg\sin{\theta}}{b} t$$

where A and B are arbitrary constants.
It is difficult to express t in term of x. But it can be solve numerically for a given x, say using Newton-Raphson method.

This is the equation for differential work, where each force in the DE is multiplied by dx. Is this perhaps a DE with an exact solution of t(xL) where xL is the ramp lenth? Starting position is x = 0. Again any help is appreciated.

$$m \frac{d^2 x}{dt^2}dx = mg\sin{\theta}dx - b \frac{dx}{dt}dx$$

I don't recall how to treat dx*dx so I'm checking my old Calculus textbook. While I'm at it, here's the power:

$$m \frac{d^2 x}{dt^2} \frac{dx}{dt} = mg\sin{\theta} \frac{dx}{dt} - b \frac{dx}{dt} \frac{dx}{dt}$$

Again I don't recall the rules for distributing differential terms at this time.

## What is "Time of Motion with Negative Feedback"?

"Time of Motion with Negative Feedback" refers to a concept in physics that describes the rate at which an object changes its position over a certain period of time while also taking into account the negative feedback that may affect its motion.

## How is "Time of Motion with Negative Feedback" calculated?

The calculation for "Time of Motion with Negative Feedback" involves dividing the change in position by the change in time while also considering the effect of negative feedback on the object's motion. This can be expressed mathematically as: Time of Motion with Negative Feedback = (Change in Position / Change in Time) x (Negative Feedback Factor).

## What is negative feedback and how does it affect motion?

Negative feedback is a term used to describe a process in which the output of a system reduces the effect of its input. In the context of "Time of Motion with Negative Feedback", negative feedback can affect the motion of an object by slowing it down or causing it to change direction, thereby influencing its overall time of motion.

## Can negative feedback ever be beneficial for an object's motion?

Yes, negative feedback can sometimes be beneficial for an object's motion. For example, in systems that require precise control and stability, negative feedback can help to maintain a steady motion and prevent overshooting or oscillations.

## What are some real-life examples of "Time of Motion with Negative Feedback"?

Some real-life examples of "Time of Motion with Negative Feedback" include the movement of a pendulum, the motion of a car's suspension system, and the behavior of a thermostat in regulating temperature. In these cases, negative feedback plays a crucial role in controlling the motion and ensuring stability.

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