When only the front wheels of an automobile are on a platform scale, the scale balances at \(8.0 \mathrm{kN} ;\) when only the rear wheels are on the scale, it balances at \(6.0 \mathrm{kN}\). What is the weight of the automobile, and how far is its center of mass behind the front axle? The distance between the axles is \(2.8 \mathrm{~m}\).

Understanding the moment balance equation is crucial for problems involving rotational equilibrium and finding the center of mass in physics. This concept is based on the principle that the sum of all moments about any point is zero when the system is in equilibrium. In the case of our automobile problem, we consider the moments produced by the forces acting on the front and rear wheels.

The moment balance equation is mathematically expressed as \( W_Ax = W_FL + W_R(L - x) \). Hier, \(W_A\) represents the total weight of the automobile, \( x \) is the distance from the center of mass to the front axle, \( L \) is the axle separation distance, \( W_F \) and \( W_R \) are the weights supported by the front and rear wheels, respectively. By selecting the front axle as the pivot point, we eliminate the moment due to the force on the front wheels, simplifying our calculations. Balancing moments allow us to solve for the unknown variable \( x \) which reflects the location of the center of mass relative to the front axle.

Force balance is another fundamental concept in physics that deals with the equilibrium of forces acting on a body. In a state of balance, the sum of all forces acting on an object is zero. Applying this to our automobile scenario, the total force balance equation is represented as \( W_A = W_F + W_R \).

This equation means that the total weight of the car, \( W_A \), is the sum of the weights on the front and rear wheels. If we simply add up the known forces of \( 8.0 \) kN and \( 6.0 \) kN from the problem statement, we can straightforwardly deduce that the car's total weight is \( 14 \) kN. Recognizing the force balance helps us not just solve for the weight, but it also reinforces our understanding of how forces interact and combine in physical systems.

Physics problem solving often involves a systematic approach to dissect complicated situations into manageable parts. By defining variables, setting up relevant equations, and employing appropriate principles, we can navigate through complex problems. This procedure is exemplified in the steps taken to solve for the automobile’s weight and the center of mass position.

In our example, we've meticulously defined variables like forces on the wheels (\(\ W_F \) and \(\ W_R \)), total weight (\(\ W_A \)), distance from the center of mass to the front axle (\(\ x \)), and the distance between axles (\(\ L \)). By setting up force balance and moment balance equations and consistently substituting known values, the unknowns are progressively uncovered. Following these steps carefully allows students to build a solid foundation for analytical thinking in physics and develop a reliable strategy for approaching a wide range of physics problems.