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To meet a U.S. Postal Service requirement, employees’ footwear must have a coefficient of static friction of 0.5 or more on a specified tile surface. A typical athletic shoe has a coefficient of static friction of 0.800. In an emergency, what is the minimum time interval in which a person starting from rest can move 3.00 m on the tile surface if she is wearing (a) footwear meeting the Postal Service minimum and (b) a typical athletic shoe?

I know how to do this, but I do not quite understand the theoretical aspects involved in it.

Why is the static force acting on the friction force during walking? How, at every step, does it act by allowing movement?

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We walk by pushing the ground backwards. At each step, we tend to push the ground backwards by our feet. There is tendency of our feet slipping backwards. In that case, friction comes into play. If enough to prevent slipping, static friction acts in forward direction on our feet, enabling us to move forward. Just imagine if there was no friction. We would never been able to 'walk'(maybe hopping like kangaroos would help!). You might have observed walking over a slippery surface gives difficulty (friction is not completely zero). A fast runner( or an athlete) has more chances of slipping, hence requires more static friction. If static friction is not enough, friction would change to kinetic thereby, causing slipping, (and even rubbing of soles! )

Now, coming to your problem. It will further clarify things:

When we talk about minimum time interval, we mean walking fast as possible, so highest tendency of slipping. So we have to consider the limiting value of static friction in the problem. So, we know the acceleration of person($a=\frac F m =\frac {\mu mg}{m}=\mu g$). We can calculate the time interval.

Hope this answer helps you!

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  • $\begingroup$ Why does "walking faster" mean having a greater tendency to slip? $\endgroup$ – gmn_1450 May 18 at 16:04
  • $\begingroup$ Can you run on a slippery surface? Suppose water/oil is spilled over ground. Will you avoid running or not? $\endgroup$ – HS Singh May 18 at 16:06
  • $\begingroup$ Faster motion is acquired by applying more force on the ground to get more reaction force and hence more acceleration $\endgroup$ – HS Singh May 18 at 16:08
  • $\begingroup$ LoL...probably not. $\endgroup$ – gmn_1450 May 18 at 16:08
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Why is the static force acting on the friction force during walking? How, at every step, does it act by allowing movement?

When we walk or run we apply a pushing force against the ground. The ground applies an equal and opposite reaction force on us. See the free body diagram of a runner below. It applies as well to a walker.

The ground reaction force on the person is resolved into the static friction force parallel to the surface and the reaction force normal to the surface. During portions of walking or running the normal reaction force is greater than the gravitational force on the person in order to lift the person off the ground.

The static friction force that the ground applies to the person propels the person forward, and is equal and opposite to the parallel force the person applies to the ground. If there were no static friction force the person will slip. Note that if the parallel component of the per pushing force on the ground exceeds the maximum possible static friction force, the person slips.

The difference between the person shown running and a person walking is the angle $θ$ is greater for the walker. When the person is standing still ($θ=90^0$) and the only forces are the persons weight and the equal and opposite normal force on the ground. There is no pushing force for friction to oppose, and so therefore no friction.

Hope this helps. enter image description here

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When a person accelerates in the forward direction, the person pushes on the ground in a backward direction and the ground pushes on the person in a forwrd direction - Newton's third law.

For a person to accelerate in the forward horizontal direction there must a force on the person in that direction - Newton's second law.

The force on the person in the horizontal forward direction is provided by the frictional force due to the ground $F$.

Assuming that there is no slipping between the shoe and the ground, the maximum magnitude of this frictional force is $F=\mu_{\rm static} W$ where $W$ is the weight of the person.

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