Classical Mechanics with Maple

Section 3.2: Balance and Conservation of Energy

Dr. Harald Kammerer
maple@jademountain.de

Initialisation

> restart;

> libname:="C:/mylib/m6dynlib","C:/mylib/m6dynfig",libname:

> with(linalg):with(plots):with(plottools):with(dynamics);with(figures_chapter_3);

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the name changecoords has been redefined

Warning, the name arrow has been redefined

3.2 Balance and Conservation of Energy

Above we distinguished active and passive forces. Additionally we separate active forces into conservative forces and non-conservative forces .

- A force is called conservative if it does no net work along a closed path. Examples are the gravitational force (see gravitational_energy ) and spring forces (see spring_energy ).

- A force is called non-conservative if it does net work along a closed path. For example the sliding (friction) force and air resistance are non-conservative.

In other words,:

- conservative systems are systems in which the total energy is conserved,

- non-conservative systems receive energy from the environment or lose energy to the environment.

Consequently we can allocate a potential to conservative forces. For example such potentials are the potential of springs and the potential of the gravity.

The result of all on a mass particle acting forces is the sum

with the passive force N which is caused by the boundary. Newton's law of motion is then

m

( Note : the forces and the acceleration in the equations are vectors)

Now we consider the motion of a mass particle along the trajectory of its motion between the two points (1) and (2) (see Fig. 6)

> display(Fig_3_4(),scaling=constrained,axes=none,title="Figure 6");

Multiplication of the law of motion with the incremental displacement d r and integration yields the equation

with the result of the conservative force F _c and the result of the non-conservative forces F _nc .

On the right side is the sum of the work of all acting forces. The passive force N is orthogonal to the trajectory and thus contributes no work. For the conservative forces there exist a total potential V ( r ), which is defined as negative work. We can say: The potential is the capability to do work . The work that is done by the conservative forces on the way from (1) to (2) is the potential difference

or with V( r1 )= V1 and V( r2 )= V2

The calculation of the work of the non-conservative forces

on the way form (1) to (2) requires that we know the force law. Further we need for the integration the form

So the motion r ( t ) must be known . In the simple case of sliding the non-conservative force is constant and the integral can be easily calculated.

We transform the left side of the equation above to

and consequently

with v 1 = v ( r 1 ) and v 2 = v ( r 2 ).

We know and at last we get

The expression

is called the kinetic energy (This relation is included in the dynamics package. See kinetic_energy )

Insert all expressions and we get the balance of energy equation

In words we say:

The sum of the kinetic and the potential energy at point (2) is the sum of the kinetic and the potential energy at point (1) increased or decreased by the work of the nonconservative forces. We get an increase of energy when the force points in the direction of the displacement and a decrease when the force points against the displacement.

In the special case that no non-conservative forces are acting we get the conservation of energy

Example: Car Braking Downhill

Have a look at a car with the mass m which is driving downwards a hill as shown in Fig. 7

> display(Fig_3_5(),scaling=constrained,axes=none,title="Figure 7");

The car drives with the constant velocity v0 in the direction of the street. The slope of the road is given by a . When the car reaches the position s=0 the driver begins to stop the car. Then the constant sliding force R takes effect parallel to the trajectory of the motion, or in other words against the coordinate s . The coordinate z points in the vertical direction and we choose z=0 at the center of gravity of the car when s=0 .

The question is: What is the position s2 of the car when it stops?

We first consider the work that is done by the sliding force R . The force R acts on the way from s1 to s2 . We can write

for the work which is done

>

Notice that R acts against the direction of the motion, so we have the negative sign.

Next we consider the potential of gravity. This potential is defined by

>

(This relation is included in the dynamics package. See gravitational_energy )

Where z is measured from an arbitrary choosen horizontal. We had set z=0 at the beginning when s=s1 . So we get

>

At the beginning of the braking process we get for the potential of gravity

> V1:=gravitational_energy(m,g,z1);

The altitude with respect to the initial height of the car when it reaches the unknown position s=s2 is

>

With this relation we get for the potential of gravity

> V2:=gravitational_energy(m,g,z2);

At last we look at the kinetic energy. At the beginning of the braking process the car drives with the velocity v0 and we get for the kinetic energy

> T1:=kinetic_energy(m,v0);

When the car reaches s=s2 it stops and the velocity is

>

So we get for the kinetic energy

> T2:=kinetic_energy(m,v2);

We write for the balance of energy

>

Isolation of s2 in this relation yields the braking distance

> s2:=solve(Energy_Balance,s2);

At last we want to calculate the braking distance for the case of a horizontal road. Then we get

>

Additionally we limit our consideration to the case that all wheels are blocking. So we have the situation of sliding friction. That yields for the braking force

>

with the coefficient of friction m . The braking distance is then

> eval(s2);

We see that here the braking distance is independent of the mass of the car.

The constant of gravity is known as

>

For example we get with the initial velocity of 75 km/h, which means in m/s

>

and the coefficient of friction

>

the braking distance (in m)

> evalf(s2);

When the car drives with the velocity of 150 km/h

>

we get

> evalf(s2);

This means that doubling the velocity yields a quadrupling of the braking distance.

Example: The Mathematical Pendulum, Energy Considerations

Now we consider the motion of the mathematical pendulum again (see above ).

> unassign('t','a','m','L','T','g','v0','omega','v','phi0','phi_d_0','phi(t)'):

> display(Fig_3_6(),scaling=constrained,axes=none,title="Figure 8");

We consider two different states. The first one is the initial state. The pendulum is displaced by f 0 and has initial angular velocity phi_d_0 . Second we consider an arbitrary state with the displacement f and the angular velocity w .

Next we have to define a horizontal from which we measure the coordinate z to describe the current height of the mass m . We choose the lowest point of the mass, i.e. f =0 . So we get for the height of the mass (see Fig. 8)

>

for the initial state and for the arbitrary second state

>

Now we get the potential of gravity for both states

> V0:=gravitational_energy(m,g,z0);

> V:=gravitational_energy(m,g,z);

The velocity of the mass particle points in the direction of the trajectory. We get for both states

>

>

and for the kinetic energy

> T0:=kinetic_energy(m,v0);

> T:=kinetic_energy(m,v);

In this example we have no non-conservative forces. So we write for the conservation of energy

> Energy_conservation:=T+V=T0+V0;

The term is on both sides of the equation and can be eliminated

> EC:=expand(lhs(Energy_conservation)-m*g*L)=expand(rhs(Energy_conservation)-m*g*L);

Have again a look at the equation of motion which we derived above, multiplied with L and m

> EOM:=g*sin(phi(t))/L+diff(phi(t),`$`(t,2)) = 0;

Multiplication with , m and yields

> NEOM:=expand(L**2*m*EOM*diff(phi(t),t));

After integration with respect to time we get

> E1:=int(lhs(NEOM),t);

We see that E1 and E0 are the same with . We say: "The conservation of energy is a first integral of the equation of motion". The integration in the last step yields an integration constant E1=E0 which represents the total energy of the system at every time. It is given by the initial conditions of the motion:

> E0:=rhs(EC);

With the conservation of energy we can give an expression for the velocity of the mass particle for every position. For example we take the same concrete values as above.

>

>

>

>

>

The conservation of energy is then

> Energy_conservation;

Solve this relation with respect to the angular velocity yields

> sol:=solve(Energy_conservation,phi_d);

We look at the graphical presentation of this solution (Fig. 9)

> P1:=plot(sol[1],phi=-0.1..0.1, color=green):

> P2:=plot(sol[2],phi=-0.1..0.1,color=red):

> display({P1,P2},title="Figure 9", scaling=constrained);

In this diagram we get for every initial condition concentric curves with the common center (0,0). This kind of diagram is called a phase portrait of the undamped system. One discrete curve is called the phase curve. Notice that the example represents a nonlinear equation of motion, because we hadn't made here the simplification . In the case of linear motions, the phase curves are ellipses. Phase portraits are very useful tools for comparing different types of vibrations.

>