Answer to Question #100523 in Classical Mechanics for Noah Samuel

Question

Compare and contrast the use of D’Alembert Principles with the principle of conservation of energy to solve an engineering Problem.
A vehicle of mass 4000kg accelerates up a road that has a slope of 1 in 10, increasing its speed from 2 to 10 m/s while travelling up the road a distance of 200m, against a frictional resistance of 5kN.

Determine:
(a) a tractive force between the driving wheel and road surface.
(b) the work done during the period of the acceleration
(c) the average power developed

Your task is:

1. Find above (a), (b) and (c) by using D’Alembert principles
2. Find above (a) (b) and (c) by using principle of ‘Conservation of Energy’
3. Compare and contrast both methods by observing you results and comments which method is best suited to solve the example and why? (Minimum 200 words).

Answer 1

The slope is

$$\theta=\text{ atan}\frac{1}{10}=5.7^\circ.$$

1. (a) The D'Alembert principle implies adding inertial forces to reach the equilibrium. Almost like in Newton's second law, but in form $F-ma=0.$

From the figure we see that the car accelerates to the right. The forces along the slope according to D'Alembert principle will be:

$$F-f-mg\text{ sin}\theta-ma=0,\\ F=f+mg\text{ sin}\theta+ma,$$

where the acceleration can be expressed as

$$a=\frac{v_f^2-f_i^2}{2d}=\frac{10^2-2^2}{2\cdot200}=0.24\text{ m/s}^2.$$

Thus:

$$F=5000+4000(9.8\cdot\text{ sin}5.7^\circ+0.24)=9853\text{ N}.$$

(b) The work:

$$W=Fd=9853\cdot200=1970667\text{ J}.$$

(c) The average power:

$$P=Fv_{av}=9853\cdot\frac{2+10}{2}=59118\text{ W}.$$

2. Now do the same using the conservation of energy.

(b) The work done by the engine is spent to increase speed, overcome friction and to lift the car up the slope:

$$W=mv_f^2/2-mv_i^2/2+mgd\text{ sin}\theta+fd=\\ \space\\ =4000\bigg(\frac{10^2}{2}-\frac{2^2}{2}+9.8\cdot200\cdot0.1\bigg)+5000\cdot200=\\=1970667\text{ J}.$$

(a) The tractive force is just work over distance:

$$F=W/d=1970667/200=9853\text{ N}.$$

(c) The time required to accelerate the car:

$$t=\frac{d}{v_{av}}.$$

The average power is just work over time:

$$P=\frac{W}{t}=\frac{Wv_{av}}{d}=\frac{1970667\cdot(2+10)}{200\cdot2}=59120\text{ W}.$$

We see that the D'Alembert principle requires less calculations and the calculations are less complex, especially if we knew of could measure the acceleration: in that case, we do not have to use squares in our computations.