Answer to Question #165281 in Mechanical Engineering for Terence

Question

Task 2 – Compare and contrast the use of D’Alembert’s principle with the principle of conservation of energy to solve an engineering problem A motor vehicle having a mass of 800 kg is at rest on an incline of 1 in 8 when the brakes are released. The vehicle travels 30 m down the incline against a constant frictional resistance to motion of 100 N where it reaches the bottom of the slope. a) Using the principle of conservation of energy, calculate the velocity of the vehicle at the bottom of the incline. b) Using an alternative method that does not involve a consideration of energy, cacluate the velocity of the vehicle at the bottom of the incline. c) Discuss the merits of the two methods you have used for parts a) and b) of this question. Justify the use of an energy method for these types of problems.

Answer 1

The slope is

$$\theta=\text{ atan}\frac{1}{8}=7.13^\circ.$$

1. (a) The D'Alembert principle implies adding inertial forces to reach 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 the D'Alembert principle will be:

a) $EK_1+EP_1=EK_2+EP_2$

$mgh_1+\frac12m(v_1)^2=mgh_2+\frac12m(v_2)^2$

$mgh_1+0=0+\frac12m(v_2)^2$

$800*9.8*30sin7.13=\frac12*800*(v_2)^2$

$(v_2)=\sqrt{\frac{800*9.8*30sin7.13}{\frac12*800}} =8.11m/s$

b)

$$v^2=2g\text{Lsin}\theta\\ v=\sqrt{2g\text{Lsin}\theta\\} =\sqrt{2*9.8\text{*30*sin}\ 7.13\\}=8.11m/s$$

c) 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.