Net Force

by Julia Logan

Definition

A Mathematical Model

In order to calculate net force, all EXTERNAL forces acting on a system are added together. The formal mathematical definition is [math]\displaystyle{ dV = -\left(E_x*dx + E_y*dy + E_z*dz\right) }[/math], where E is the electric field with components in the x, y, and z directions. Delta x, y, and z are the components of final location minus to the components of the initial location.

A Computational Model

Net force is an essential component of the Momentum Principle! We can use the Momentum Principle in vpython to update the position of a moving object. But first, we have to find net force.

  #1 Fspring = -k*s
  #2 Fgravmag = mball * g
  #3 Fgrav = Fgravmag * vector(0,-1,0)
  #4 Fnet = Fspring+Fgrav
  #5 pball = pball + Fnet * deltat
  #6 vball = pball / mball
  #7 ball.pos=ball.pos+vball*deltat

Here, the spring force and the gravitational force are found using formulas (lines 1-3). Then, they are added together to get the net force on the object (in this case a ball, line 4). The net force is then used in the update form of the momentum principle (line 5). In line 6 the velocity is updated, and line 7 the position is updated. Without net force calculations, tracing an object's path would be impossible.

Tracing the path of a ball/spring model in vpython using net force and the momentum principle.

Simple Example

In this example, the electric field is equal to [math]\displaystyle{ E = \left(E_x, 0, 0\right) }[/math]. The initial location is A and the final location is C. In order to find the potential difference between A and C, we use [math]\displaystyle{ dV = V_C - V_A }[/math].

Since there are no y and z components of the electric field, the potential difference is [math]\displaystyle{ dV = -\left(E_x*\left(x_1 - 0\right) + 0*\left(-y_1 - 0\right) + 0*0\right) = -E_x*x_1 }[/math]

Let's say there is a location B at [math]\displaystyle{ \left(x_1, 0, 0\right) }[/math]. Now in order to find the potential difference between A and C, we need to find the potential difference between A and B and then between B and C.

The potential difference between A and B is [math]\displaystyle{ dV = V_B - V_A = -\left(E_x*\left(x_1 - 0\right) + 0*0 + 0*0\right) = -E_x*x_1 }[/math].

The potential difference between B and C is [math]\displaystyle{ dV = V_C - V_B = -\left(E_x*0 + 0*\left(-y_1 - 0\right) + 0*0\right) = 0 }[/math].

Therefore, the potential difference A and C is [math]\displaystyle{ V_C - V_A = \left(V_C - V_B\right) + \left(V_B - V_A\right) = E_x*x_1 }[/math], which is the same answer that we got when we did not use location B.

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