A 70.0 kg climber is supported in the "chimney" by the friction forces exerted on his shoes and back. The static coefficients of friction between his shoes and the wall, and between his back and the wall, are 0.760 and 0.610, respectively. What is the minimum normal force he must exert? Assume the walls are vertical and that friction forces are both at a maximum.
Well, if the climber wants to stay in place and not come crashing down, he needs to make sure the friction forces are strong enough to support his weight.
Let's start with the friction force between his shoes and the wall. The static coefficient of friction is given as 0.760. Now, we can use the formula for friction force, which is F = µN (where F is the friction force, µ is the coefficient of friction, and N is the normal force).
So, for the shoe-wall friction, the friction force would be 0.760N.
Now let's move on to the friction force between his back and the wall. Here the static coefficient of friction is 0.610. So, the friction force between his back and the wall would be 0.610N.
Now, in order for the climber to stay in place, both these friction forces have to balance out his weight which is 700N (70kg multiplied by the gravitational acceleration of 9.8m/s²).
So, we can set up the equation: 0.760N + 0.610N = 700N
Solving this equation, we find that the normal force (N) should be approximately 970.08N. The climber must exert at least this much normal force to stay put.
That's a tremendous force and a testament to the strength of our climbers. Maybe I should stick to making people laugh instead of attempting these extreme sports!
To find the minimum normal force the climber must exert, we need to analyze the forces acting on him.
First, let's consider the vertical forces acting on the climber. The only vertical force is the gravitational force, which is equal to the weight of the climber and can be calculated as:
Weight = mass x acceleration due to gravity
= 70.0 kg x 9.8 m/s²
= 686 N
Since the climber is not accelerating vertically, the normal force exerted by the wall must balance the gravitational force. Therefore, the normal force is also 686 N.
Now let's consider the horizontal forces acting on the climber. There are two friction forces: one between the shoes and the wall, and the other between the back and the wall. In both cases, the friction force is determined by the maximum static coefficient of friction and the normal force.
Friction force between shoes and wall:
Friction1 = static coefficient of friction1 x normal force
= 0.760 x 686 N
= 521.36 N
Friction force between back and wall:
Friction2 = static coefficient of friction2 x normal force
= 0.610 x 686 N
= 418.66 N
To determine the minimum normal force, we need to find the maximum of the two friction forces. In this case, the maximum friction force is 521.36 N. Therefore, the climber must exert a minimum normal force of 521.36 N to support himself in the chimney.