When only the front wheels of an automobile are on a platform scale, the scale balances at 7.8 kN; when only the rear wheels are on the scale, it balances at 5.5 kN. What is the weight of the automobile?
How far is its center of mass behind the front axle? The distance between the axles is 2.5 m.
To find the weight of the automobile, we can use the principle of conservation of weight. When only the front wheels are on the platform scale, the scale balances at 7.8 kN, which represents the weight of the front of the automobile. Similarly, when only the rear wheels are on the scale, the scale balances at 5.5 kN, representing the weight of the rear of the automobile.
Let's denote the weight of the front of the automobile as Wf and the weight of the rear as Wr. We can set up the following equations:
Wf = 7.8 kN -- Equation 1
Wr = 5.5 kN -- Equation 2
The total weight of the automobile, W, is simply the sum of the weights of the front and the rear:
W = Wf + Wr
To find the weight of the automobile, we need to substitute the values of Wf and Wr from Equations 1 and 2:
W = 7.8 kN + 5.5 kN
W = 13.3 kN
Therefore, the weight of the automobile is 13.3 kN.
Now let's move on to determining the distance of the center of mass behind the front axle. We can assume a uniform distribution of weight along the length of the automobile. The total distance between the front and rear axles is given as 2.5 m.
Let's consider the distance of the center of mass from the front axle as x meters. The moment of the weight of the front about the front axle is equal to the moment of the weight of the rear about the front axle, assuming the scale is level. Mathematically, this can be expressed as:
Wf * x = Wr * (2.5 - x)
Substituting the known values of Wf and Wr:
7.8 kN * x = 5.5 kN * (2.5 m - x)
Simplifying the equation:
7.8kN * x = 13.75 kN * m - 5.5 kN * x
13.3 kN * x = 13.75 kN * m
Dividing both sides by 13.3 kN, we find:
x = 13.75 kN * m / 13.3 kN
x ≈ 1.034 m
Therefore, the center of mass of the automobile is approximately 1.034 meters behind the front axle.