The 60 kg climber in Fig. 4-52 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 0.82 and 0.58, respectively. What is the minimum normal force he must exert? Assume the walls are vertical and that friction forces are both at a maximum.

Draw the free body diagram and use:

mg=N(μ1+μ2)
m=mass of climber
g=acceleration du to gravity
N=minimum normal force, N
μ1 and μ2 = coeff. of friction.

To find the minimum normal force the climber must exert, we need to consider the maximum friction forces that can be generated by the coefficients of friction.

Let's denote the static coefficient of friction between the climber's shoes and the wall as μs1 = 0.82, and the static coefficient of friction between the climber's back and the wall as μs2 = 0.58.

The maximum friction force that can be exerted by the climber's shoes is given by:

F1 = μs1 * N

where N is the normal force exerted by the climber on the wall.

Similarly, the maximum friction force that can be exerted by the climber's back is given by:

F2 = μs2 * N

To determine the minimum normal force, we need to find the maximum value between F1 and F2.

F1 is the limiting factor when:

F1 ≥ F2

Substituting the expressions for F1 and F2, we have:

μs1 * N ≥ μs2 * N

Simplifying, we get:

0.82 * N ≥ 0.58 * N

Now, to find the minimum normal force, we want to solve for N.

Dividing both sides of the inequality by N, we get:

0.82 ≥ 0.58

Since this inequality is true, it means that the climber's shoes provide the maximum friction force needed to keep the climber supported in the chimney. Therefore, the climber's shoes are the limiting factor, and the minimum normal force the climber must exert is:

N = F1 / μs1

Substituting the values we know:

N = F1 / 0.82

Since we don't have the value of F1, it is not possible to calculate the minimum normal force with the given information.

To find the minimum normal force that the climber must exert, we need to consider the maximum static friction forces between his shoes and the wall, and between his back and the wall.

The formula for the maximum static friction force is given by:

F_friction = μ * N

where F_friction is the friction force, μ is the coefficient of friction, and N is the normal force.

In this case, we have two coefficients of friction: μ_shoes = 0.82 and μ_back = 0.58.

To calculate the minimum normal force, we need to find the maximum friction forces for both his shoes and back.

For the shoes:

F_friction_shoes = μ_shoes * N

And for the back:

F_friction_back = μ_back * N

Since the friction forces are both at a maximum, we can equate the two forces:

F_friction_shoes = F_friction_back

μ_shoes * N = μ_back * N

Since the normal force cancels out, we can solve for N:

N = F_friction_shoes / μ_shoes

N = F_friction_back / μ_back

Substituting the given values of the coefficients of friction:

N = F_friction_shoes / 0.82

N = F_friction_back / 0.58

To find the maximum friction forces, we need to consider the weight of the climber. The weight is given by:

Weight = m * g

where m is the mass of the climber and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Substituting the given mass of the climber:

Weight = 60 kg * 9.8 m/s^2

Now we can find the maximum friction forces:

F_friction_shoes = μ_shoes * Weight

F_friction_back = μ_back * Weight

Substituting the given values of the coefficients of friction:

F_friction_shoes = 0.82 * (60 kg * 9.8 m/s^2)

F_friction_back = 0.58 * (60 kg * 9.8 m/s^2)

Calculating these values will give us the maximum friction forces for the shoes and back.

Finally, substitute these values into the equations for N:

N = F_friction_shoes / 0.82

N = F_friction_back / 0.58

Calculating these values will give us the minimum normal force the climber must exert.