• Initialisation
• 3.1 Newton's Laws
  • Example: The Mathematical Pendulum
  • General Strategy to Get The Equation of Motion

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3.1 Newton's Laws

Newton's second law gets into a quantitative relationship between force and motion. In the form of an equation, the second law reads

(As in section 2, the derivative of a variable with respect to time is ususly denoted by a dot above the variable. Here it is denoted by an appended letter d. So we write for the coordinates of the acceleration in x-direction: xdd . Same is valid for the components in y- and z-direction.)

Desscriptions in polar or natural coordinates are also possible.

Newton's third law addresses the nature of forces. This third law states that the force resulting from the interaction of two bodies acts with equal magnitude on both of them and in the opposite directions. For every action, there is an equal and opposite reaction.

Forces that act on the particle due to geometrical boundaries that restrict the particle's motion are called passive forces .

Forces that are completely described by a force law are called active forces . Such forces are for example the weight force, the spring force and the sliding (friction) force.

Example: The Mathematical Pendulum

Now we consider the mass particle in a displaced position f . The trajectory of the motion is known as a circle with radius L . From the kinematics in section 2 we know for the acceleration in polar coordinates

> a:=polar_acc(L,phi(t),t);

The radial component of the acceleration points from the mass particle to the suspension point. The inertial force Z belonging to it acts in the opposite direction, away from the suspension point. We write

> Z:=-m*a[1];

The tangential component of the acceleration points in the direction e f (ephi in the figures). Thus the inertial force belonging to it acts in the opposite direction. We write

> T:=-m*a[2];

In Fig. 2 we see all acting forces on the mass particle.

Notice: In the figure the forces Z and T are drawn in the positive coordinate direction. In reality the force T shows in the opposite direction as drawn in the figure, but this is respected by the negative sign in the definition of T .

> display(Fig_3_2(),scaling=constrained,axes=none,title="Figure 2");

The equilibrium in tangential and radial direction yields

>

>

The force law for the weight force is

>

G always points downwards. This is taken into account in Fig 2, so there is no negative sign in the force law.

Substituting the force law into both equilibrium conditions yields

> EQt:=subs(FLG,EQt);

> EQr:=subs(FLG,EQr);

For and we get from EQt the equation of motion for the system

> EOM:=expand(-EQt/(m*L));

This is an ordinary differential equation of degree 2 with respect to f ( t ).

From EQr we get the force S in the thread as

> S:=solve(EQr,S);

After solving the equation of motion we can calculate the force S in dependence of the time t .

The derived equation of motion is nonlinear and cannot be easily solved. But for small angles f we can linarize the equation.

For f << 1 we make the substitutions

>

>

One more substitution is usual. We write

or in equivalent but more practicable manner in MAPLE

>

So we get the linearized equation of motion

> LEOM:=subs({subst1,subst2,subst3},EOM);

Now we get for the solution of LEOM

> sol:=dsolve(LEOM,phi(t));

The integration constants _C1 and _C2 depend on the initial conditions.

At time t=0 the angle f (t) should be f0

> IC1:=phi(0)=phi0;

and the angular velocity diff( f (t),t) should be f_ d _0

> IC2:=D(phi)(0)=phi_d_0;

So we get for the complete solution

> Csol:=dsolve({LEOM,IC1,IC2},phi(t));

The expression w is called natural angle frequency . It is an important measure to describe periodic motions. The unit of w is 1/s (1 per second).

Sometimes one uses the natural frequency

to describe the motion. f is the number of cycles of a motion within one second. The unit of f is 1 Hz (Hertz).

The reciprocal value of f is called the period.

or

In our example the period is

> T:=expand(subs(1/omega=sqrt(L/g),2*Pi/omega));

Here we make the third substitution subst3 back.

We see that in this system the periodic time is independent of the mass and the initial conditions.

Finally we consider the pendulum's motion with some concrete values. We know for the acceleration of gravity

>

The following values for the mass and the length of the thread can be changed. But don't forget that the solution is only valid for angles f << 1.

>

>

So we get for the motion

> assign(subs(1/omega=sqrt(L/g),Csol)):

>

The natural angle frequency of the motion is

>

and the natural frequency is

>

The period of the motion is

> evalf(T);

In Fig. 3 we see the time history of the motion over 10 periods for the following initial conditions

>

>

> P1:=plot(phi(t),t=0..10*T, color=blue):

> display(P1,title="Figure 3");

Fig. 4 shows the time history of the force S .

> P2:=plot(S,t=0..10*T, 0..m*g*1.2, color=black):

> display(P2,title="Figure 4");

We see that the force in the thread is always positive. Further we see that the frequency of the thread force S is two times the frequency of the angle f . We don't discuss this phenomen further at this point because it is not subject of this course.

Fig. 5 shows a little animation of the motion of the pendulum. You can try what happens when you change the initial conditions or the mass or the length of the pendulum. To start the animation click on the picture and the "play" in the menu bar.

> display(Fig_3_3(L,m,eval(phi(t)),t,10*T),insequence=true,scaling=constrained,axes=none,title="Figure 5");