Problem 34 - An Introduction to the Mechanics of Solids in SI Units

Question

The clean-air car shown has the following characteristics: Wheelbase \(L=250 \mathrm{~cm}\) Weight \(W=10 \mathrm{kN}\) Weight distribution (on level ground), 60 percent on rear wheels Power h.p. \(=75 \mathrm{MW}\), rear-wheel drive Height of center of gravity \(h=0.5 \mathrm{~m}\) Wheel diameter \(d=0.5 \mathrm{~m}\) If the coefficient of friction between tires and road is \(f=0.7\), what is the maximum hill angle \(\theta\) that can be climbed?

Step-by-Step Solution

Answer

The maximum hill angle that the car can climb is given by \(\theta_{max} = \arcsin (\frac{4.2}{6})\).

Step 1: Determine the weight on the rear wheels

The weight distribution on the rear wheels is given to be 60%. Hence, the weight on the rear wheels, \(W_r\), can be calculated by taking 60% of the total weight (\(W\)), i.e., \(W_r = 0.60 * W = 0.60 * 10 \mathrm{~kN} = 6 \mathrm{~kN}\).

Step 2: Calculate the maximum force of friction

The maximum force of friction between the road and the car's tires, \(F_f\), that can be exerted without the car sliding back can be calculated using the formula: \(F_f = f * N\), where \(f\) is the coefficient of friction and \(N\) is the normal force, which is the weight on the rear wheels. Hence, \(F_f = f * W_r = 0.7 * 6 \mathrm{~kN} = 4.2 \mathrm{~kN}\).

Step 3: Determine the maximum component of vehicle weight downhill

The maximum component of the vehicle weight that can act downhill, \(W_{dmax}\), is equal to the maximum force of friction, which is 4.2 kN.

Step 4: Calculate the maximum hill angle

The maximum angle (\(\theta_{max}\)) can be derived from the trigonometric relation: \(\sin \theta = \frac{W_{dmax}}{W_{r}} = \frac{4.2}{6}\). Solving this gives \(\theta_{max} = \arcsin (\frac{4.2}{6})\).

Key Concepts

Coefficient of Friction

Understanding the coefficient of friction is crucial when analyzing any situation involving movement and contact between surfaces. Essentially, it is a measure that reflects the grip between two surfaces – in this case, a vehicle's tires and the road surface. Represented by the symbol \(f\), the coefficient of friction is a dimensionless number that ranges from 0 to 1, where 0 means no friction (imagine ice) and 1 represents maximum friction.

An essential point to note is that the coefficient of friction is determined by both the materials in contact and the condition of their surfaces. A highly textured tire on a rough road would have a higher \(f\) value compared to a worn tire on a slick, wet road. For our problem, a coefficient of friction of 0.7 indicates a fairly good grip, which is necessary for climbing steep hills.

Weight Distribution

The concept of weight distribution pertains to how the total weight of an object, such as a car, is shared across its support points – in this scenario, the wheels. A car with a weight distribution of 60 percent on rear wheels suggests that a more significant portion of the car's weight is supported by the rear. This distribution turns out to be fundamental when calculating traction, as more weight over the driving wheels generally means more grip, all else being equal.

In our calculation, weight distribution directly affects the normal force, which in turn influences the force of friction that can be achieved. Therefore, when climbing a hill, optimal weight distribution towards the driving wheels (rear wheels for this car) is advantageous, helping maximize the vehicle's ability to ascend without slipping backwards.

Wheelbase

The wheelbase of a vehicle is the distance between the centers of the front and rear wheels, denoted here as \(L\). This dimension is significant for several reasons, including stability, handling, and weight distribution. A longer wheelbase tends to provide a more stable ride, which is beneficial for maintaining control, especially on inclines.

However, for our hill climbing problem, the primary concern with wheelbase is related to the car's ability to maintain contact with the ground over uneven terrain. A short wheelbase might make it easier for a vehicle to navigate steep changes in slope without tipping, but it could also mean less weight is distributed over the rear wheels, potentially reducing traction. On the other hand, a longer wheelbase, like the one we have here, may distribute weight more evenly, which can be an advantage unless the hill's gradient changes abruptly.

Center of Gravity

The center of gravity (CG) is the point at which an object's weight is considered to be concentrated. For stability, it's preferable for the CG to be as low as possible, especially in vehicles. Having a CG at a height of 0.5 meters, our car's tendency to tip over becomes an important aspect when dealing with hill angles.

While the center of gravity does not directly factor into the calculations for maximum hill angle, it's an underlying aspect that affects the car's dynamics. A lower CG would lessen the risk of the car tipping backward on a steep climb, due to a more balanced weight distribution along the vertical axis. Hence, the placement of the center of gravity significantly influences vehicle stability and assists in smoother navigation of inclines.