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## Physics Tutorials

### Friction Force Limits a Car's Acceleration on the Road

Definition: The friction law states that the sliding friction force is proportional to the perpendicular reaction of the bearing surface and opposite to the body's movement direction. The coefficient of friction, μ, depends solely upon the properties of the materials rubbing. The rest friction force appears whenever there is a force acting upon the resting body, and it is of equal magnitude and opposite direction. When the maximum rest friction force is exceeded, sliding begins.

#### Explanation

This means that whenever you apply a small force to a body resting on a rough plane, the rest friction force will be annihilating the former (according to the Third Newton's Law). For example, if you are trying to push forward a box that is too heavy, the rest friction will be holding it at the initial position motionless. But as long as you increase your force so that it will exceed the maximum rest friction, the box will start moving away from the spot. The maximum rest friction force is usually a bit bigger in magnitude than the sliding friction force. This is why it is always a little harder to make a resting box move than to keep it moving.

#### Applications:

A car's engine makes its wheels move and each part of the wheel that touches the ground moves in the opposite direction. This causes the friction force from the road surface to appear and act to the direction of the car's movement. Thus it appears that the acting force in the car movement is the friction between the wheels and the ground.

#### Question

The sliding coefficient of friction between the car wheels and the road is μ =.5. What is the largest acceleration that the car may achieve moving straight on a horizontal road? Presume that the engine power [and mass are equally distributed to all four tires].

SOLUTION: As it has already been explained, the acting force F that moves the car is the friction between the wheels and the road. Let a be the acceleration, so according to the Second Newton's Law, F=m*a where m is the mass of the car. It can be seen that the acceleration will reach its maximum when the force F will do the same, and it will happen when the wheels will start sliding. In this case the friction force will be F= μ*N, where N is the perpendicular reaction force of the road; as the car does not move vertically, it equals the car's weight, m*g (g is the free fall acceleration, it is almost 10 m/s/s).

So we get F=μ*m*g and F=m*a, that means μ*m*g=m*a and a=μ*g.
In our case the maximum acceleration possible will be (.5)*10=5 m/s/s.

[If power is delivered to one tire bearing one quarter the weight of the car, then maximum acceleration will be

F=μ*(0.25)*m*g and F=m*a, that means μ*(0.25)*m*g=m*a and a=μ*g/4

So the maximum acceleration would be (.5)*(10)/4 = 1.25 m/s/s)

This explains the advantage of a well balanced 4-wheel drive vehicle.]

We see that the acceleration depends on the coefficient of friction [and weight and power distribution], no matter how strong the engine is. [The coefficient of friction of rubber on wet ice may be as low as .05.] That is why a car can hardly [accelerate] on slippery ice.