# I Force as a function of time/space

1. Nov 19, 2018

### fog37

Hello,

While standing somewhere, we can accelerate a RC car more or less using the remote control toggle. The electric motor then applies a varying force $F$ to the car's wheels depending on when we act on the toggle. Does that varying force represent an example of a time-dependent force $F(t)$ since it varies according to the time we act on the toggle? However, from the car's perspective, the force varies with time but the car occupies a certain position in time, i.e. $t(x)$, so the force can be implicitly made to depend on the position coordinate $x$ of the car as well....

As another example, a mass attached to a horizontal, oscillating spring, experiences a force $F(x)=-kx$ which depends on the spring position. Everytime the mass is the position $x$, the force is the same, regardless of when it is there, so the force is not dependent on time. However, the mass position is a function of time, $x(t)$, so we could write the spring force $F(x)$ as function of t and all of a sudden the spring force becomes a function of time: $F(t)$. So, should we state that the spring force is a position-dependent force in an absolute sense or is it equally a time-dependent force since the force, from the oscillating mass perspective, is different at different time instants?

2. Nov 19, 2018

### gmax137

You can parameterize the equations in terms of time or position. Sometimes one or the other choice is much more convenient or easier to solve.

3. Nov 19, 2018

### fog37

Sure, but certain forces are position dependent, some time-dependent and some position and time dependent...The spring force is termed a position-dependent force and not a time-dependent force: $F_{spring}= - k x(t)$ which is explicitly a function of $x$ and implicitly a function of $t$....

4. Nov 19, 2018

### Staff: Mentor

It doesnâ€™t depend on time. If you have the spring at the same place you have the same force regardless of whether it is there now or later.

5. Nov 19, 2018

### A.T.

In physics "A is a function of B" means that a change of B always implies a change of A, not just in some examples.

6. Nov 20, 2018

### PeroK

There is a general point here that comes up quite often. If we have a spring or a gravitational or electric field, then these systems may be time-independent. The spring is the same spring today, tomorrow and the day after; the gravitational field may vary across space but not be changing with time; and, the same for the electric field.

On the other hand, we may have a spring that is cooling down, heating up or wearing out and its spring constant changing. Or, we may have gravitational filed that is changing with time etc.

Now, a very different concept of "changing with time" or being a "function of time" comes when we look at the motion of a particle in these systems. A particle then has a trajectory - sinusoidal in SHM, or elliptical in a gravitational field etc. This trajectory is then a function of time: $\vec{r}(t)$. And the force experienced by that particular particle is then also a function of time, as it moves through the potential field.

7. Nov 20, 2018

### BilbobagginsINSPACE

I don't believe so. Let's say we tell the car to apply 20 Newtons of force. at 2 seconds, it would measure 20 Newtons, and at 5 seconds in, it would still have 20 Newtons. It's velocity and acceleration as a result of that force will change, but the force it self won't.

8. Nov 20, 2018

### Staff: Mentor

Are you familiar with Newton's Second Law, F=ma. If force is constant, then acceleration is constant, not changing.

9. Nov 20, 2018

### PeroK

Do Newton's laws apply in Middle Earth?

10. Nov 20, 2018

### A.T.

How do you that? You have no direct dial for the net force in a car.

11. Nov 20, 2018

### fog37

Thank you for the inputs.

One more clarification: let's consider a force field that is constant (in time) but varies with position. An object immersed in this field that is free to move will move, it will be changing its positions and will be experiencing, from its perspective, a time-varying force. But the force itself is not time-varying.

The earth's gravitational field is constant in time but varies with position. A falling object experiences, from its own perspective, an increasing force that increases with time, not because the field is changing in time, but because the object experiences a larger and larger gravitational force as it descends and occupies new and lower position.

So, when we state that a force or a force field is either time-dependent or position dependent, we should never look at the force structure not from the perspective of the moving object otherwise all forces will appear to be both time and position dependent.

12. Nov 20, 2018

### Staff: Mentor

Many problems in mechanics are "frame dependent". For example, all motion is relative. We can always choose a frame in which uniform motion is not moving at all.

13. Nov 23, 2018

### fog37

Hello everyone. I think I finally clear my doubts on this topic. Let's see if you agree:

The most general form of a force F can be a function $F(x,v,t)$. In the 1D case, the the force function $F(x,t,v)$ and the object's initial conditions IC must be provided before we can solve the differential equation $ma= m \frac {dx^2}{dt^2} = F(x,t,v)$ to find the equation $x(t)$ describing the object's position in time. From $x(t)$, we can take the derivatives of $x(t)$ to find $v(t)$ and $a(t)$.

When the force $F(x)$ only varies with position, it means that it is not time-varying and its value is contact in time at any fixed spatial position $x$. But a hypothetical object moving under the influence of $F(x)$ will be changing its positions in time and will be experiencing, from its own perspective, a time-varying acceleration $a(t)$, hence a time-varying force, since the force the object experiences changes as time goes by. For example, The earth's gravitational field is constant in time but varies with position and a falling object experiences a force that increases with time, not because the field is changing in time, but because the object experiences a larger and larger gravitational force as it descends and occupies new and lower position.

Here another example: in the case of linear air resistance $F = - k v$, the acceleration depends solely on the object's speed $v$. But the object has a certain speed $v$ at different time instants t, i.e. $v(t)$. But we can also parametrize $v$ in terms of the object's position $x$ to obtain $v(x)$. If we know $v(t)$ and $v(x)$, we could plug either into $F=-kv$ and obtain a force function $F$ as $F(t)$ or even as $F(x)$. This way we would find the exact and "specific" force that the object experiences either in time or space. But those force equations $F$ would be different for different objects depending on the object's initial conditions: different ICs would determine different $v(t)$ and $v(x)$ functions which would make $F$ different for every different set of ICs. This means that the force structure would not appear constantly equal to F=-kv and vary depending on the specific object and its specific ICs. That loss of generality does not capture the real structure of the force making the equation that describes the force dependent on the object's initial conditions ICs...