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 need to add the forces acting on the front and rear wheels.

Let's assume that the weight of the automobile is W (in kilonewtons).

When only the front wheels are on the scale, the scale balance is 7.8 kN. This means that the force exerted by the front wheels is equal to the weight of the automobile plus the force exerted by the rear wheels.

So, the force exerted by the front wheels = W + force exerted by the rear wheels.

Similarly, when only the rear wheels are on the scale, the scale balance is 5.5 kN. This means that the force exerted by the rear wheels is equal to the weight of the automobile plus the force exerted by the front wheels.

So, the force exerted by the rear wheels = W + force exerted by the front wheels.

From the given information, we have the following equations:

W + force exerted by the rear wheels = 7.8 kN ---- Equation 1

W + force exerted by the front wheels = 5.5 kN ---- Equation 2

Subtracting Equation 2 from Equation 1, we get:

force exerted by the rear wheels - force exerted by the front wheels = 7.8 kN - 5.5 kN

Simplifying further, we have:

force exerted by the rear wheels - force exerted by the front wheels = 2.3 kN

Substituting the expressions for the forces exerted by the wheels:

W + W = 2.3 kN

2W = 2.3 kN

Dividing both sides of the equation by 2, we get:

W = 2.3 kN / 2

W = 1.15 kN

Therefore, the weight of the automobile is 1.15 kN.

To find the distance of the center of mass behind the front axle, we can use the concept of a lever arm.

The lever arm is the perpendicular distance between the point of force and the axis of rotation. In this case, the axis of rotation can be considered as the front axle.

Let's assume that the distance of the center of mass behind the front axle is D meters.

Since the weight of the automobile acts downwards, it will create a clockwise torque around the front axle. The torque can be calculated by multiplying the weight of the automobile by the lever arm.

Torque = Weight * Lever Arm

The torque created by the weight of the automobile is equal to the torque created by the force exerted by the rear wheels.

So, (Weight of the automobile) * D = (Force exerted by the rear wheels) * (Distance between the axles)

Substituting the values we know, we have:

1.15 kN * D = 5.5 kN * 2.5 m

Simplifying further, we get:

D = (5.5 kN * 2.5 m) / 1.15 kN

D ≈ 11.96 m

Therefore, the distance of the center of mass behind the front axle is approximately 11.96 meters.