Weight of an automobile, whilst only knowing tire pressure.?

Question

The four tires of an automobile are inflated to a gauge pressure of 218000 Pa. Each tire has an area of 0.0297 m^2 in contact with the ground.
A. Determine the weight of the automobile. Answer in units of N.

Answer 1

Pressure = Force / Area

You know the pressure and the area; find the force. That force is a quarter of the cars weight (area is per tire and you have four tires).

Answer 2

This is an equal but opposite force problem. So first thing we always do with a force problem is ID all the forces at play.

First there is the force of gravity...aka weight W = mg; where m is the mass of the car and g is acceleration due to gravity.

Is that all there is? If gravity is the only force and gravity always acts vertically downward, toward the center of the Earth, then the car would be accelerated at g = 9.81 m/sec^2 into the center of Earth. But it's not accelerating is it.

Therefore there has to be an equal but opposite force F offseting the force of gravity and giving us a net force f = 0 Newtons. In math talk, that's f = 0 = ma = W - F. Where a is the car's acceleration up and down (since the forces are vertical forces) and a = 0 because the car is not moving.

So there we have it W = F and we can find the weight of the car if we find F. Where does F come from? F is the force of the ground the car sits on pushing up on four tires.

Assuming the weight of the car is evenly distributed over the four tires, we can write F = 4f; where f = pA is the force on each tire with pressure p = 218000 N/m^2 over contact area A = .0297 m^2. Then W = F = 4f = 4 pA = the weight of the car in Newtons. You can do the math.

The physics is this. When a mass (car) is stationary, all the forces acting on it add up to zero net forces f = 0. This results because when v = 0 and the object is not moving, which is a constant velocity of zero, dv/dt = a = 0 so that f = ma = m0 = 0. A stationary mass must have net forces f = 0. QED.

PS: I don't believe atmospheric pressure has any bearing on this question. The force F that opposes the weight W is the net force gotten from the net pressure and that's what we read in the air pressure gauge. That is, the gauge pressure already accounts for atmospheric pressure; to fold that into F would be to double count its effect.

Answer 3

Just use the definition of pressure (force divided by the area over which it acts) in reverse. All you have to do is multiply the pressure in each tire by the area of ground the tire is touching and add it up for all four tires.

This really works for a real car. If you shove paper under the tire as far as you can from each side to measure how much tire area is in contact with the ground and also measure the pressure with a tire gauge, you can find the weight of the car. Try it!

Answer 4

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Absolute pressure in tire=gauge pressure ( 218000 Pa)+atmospheric pressure(101000 Pa)=319000 Pa

Absolute pressure in one tire=319000 Pa

force on one tire= pressure * area =319000* 0.0297

force on one tire=37897.2 N

force on four tires = 37897.2 N

the weight of the automobile=force on four tires

the weight of the automobile is 37897.2 N
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