Shoes made for the sports of bouldering and rock climbing are designed to provide a great deal of friction between the foot and the surface of the ground. Such shoes on smooth rock might have a coefficient of static friction of 1.2 and a coefficient of kinetic friction of 0.90.

For a person wearing these shoes, what’s the maximum angle (with respect to the horizontal) of a smooth rock that can be walked on without slipping?

50

To determine the maximum angle at which a person wearing these shoes can walk on a smooth rock without slipping, we'll use the coefficient of static friction. The maximum angle can be found using the formula:

tan(θ) = coefficient of static friction

First, we need to calculate the maximum angle in terms of the coefficient of static friction:

θ = tan^(-1) (coefficient of static friction)

θ = tan^(-1) (1.2)

Using a scientific calculator, we find:

θ ≈ 50.19 degrees

Therefore, a person wearing shoes designed for bouldering and rock climbing can walk on a smooth rock without slipping up to a maximum angle of approximately 50.19 degrees with respect to the horizontal.

To find the maximum angle at which a person wearing these shoes can walk on a smooth rock without slipping, we can use the concept of maximum angle of inclination or the angle at which the force of gravity is equal to the maximum frictional force.

First, let's calculate the maximum frictional force. The maximum static friction force (F_max) between the shoe and the rock can be found using the formula:

F_max = coefficient of static friction * normal force

The normal force is the force exerted by the rock on the shoe, which is equal to the weight of the person (mg), where m is the mass and g is the acceleration due to gravity.

Now, let's assume the weight of the person is W.

F_max = coefficient of static friction * W

Next, the maximum angle (θ_max) can be calculated using the following trigonometric relationship:

F_max = W * sin(θ_max)

Rearranging the equation, we get:

sin(θ_max) = coefficient of static friction

Finally, we can find the maximum angle by taking the inverse sine (arcsin) of the coefficient of static friction:

θ_max = arcsin(coefficient of static friction)

For the given coefficient of static friction of 1.2, we can calculate:

θ_max = arcsin(1.2)

However, the maximum value for the sine function is 1. Therefore, if the coefficient of static friction is greater than 1, it means the shoe will slip even on a completely horizontal surface. So, in this case, the given coefficient of static friction of 1.2 is not physically possible, and we need to consider the coefficient of kinetic friction instead.

The coefficient of kinetic friction is denoted as μ_k, which is 0.90 in this case. Using the same calculations, we can find the maximum angle as:

θ_max = arcsin(0.90)

Now, let's plug in the values and calculate:

θ_max = arcsin(0.90)
θ_max ≈ 64.5 degrees

Therefore, a person wearing these shoes can walk on a smooth rock without slipping up to a maximum angle of approximately 64.5 degrees with respect to the horizontal.