Consider the 52.0-kg mountain climber in
the figure to the right. Find (a) the tension
in the rope and (b) the force that the
mountain climber must exert with her feet
on the vertical rock face to remain
stationary. Assume that the force is exerted
parallel to her legs. Also, assume negligible
force exerted by her arms. (c) What is the minimum coefficient
of friction between her shoes and the cliff
A=512.1N
B=273.0N
C=coefficient of friction is .268
In order to find the tension in the rope and the force the mountain climber must exert with her feet on the vertical rock face, we need to consider the forces acting on her.
Let's label the tension in the rope as T, the normal force exerted by the cliff as N, and the force the climber exerts with her feet as F.
(a) To find the tension in the rope (T), we can use the equation for vertical equilibrium:
ΣFy = 0
In this case, the only vertical force acting on the climber is the gravitational force, which is given by:
Fg = mg
where m is the mass of the climber and g is the acceleration due to gravity (approximately 9.8 m/s^2).
So, ΣFy = T - Fg = 0
T = Fg
T = (52.0 kg)(9.8 m/s^2)
T = 509.6 N
Therefore, the tension in the rope is 509.6 N.
(b) To find the force the climber must exert with her feet (F) to remain stationary, we can consider the horizontal equilibrium of forces:
ΣFx = 0
In this case, the only horizontal force acting on the climber is the friction force, which can be written as:
Ff = μN
where μ is the coefficient of friction and N is the normal force.
Since the climber needs to remain stationary, the friction force should be equal to the force she exerts with her feet:
Ff = F
So, F = μN
Now we need to calculate the normal force N. Since the climber is not accelerating vertically, the normal force should be equal to the gravitational force acting on her:
N = mg
N = (52.0 kg)(9.8 m/s^2)
N = 509.6 N
Therefore, the force the climber must exert with her feet to remain stationary is also 509.6 N.
(c) To find the minimum coefficient of friction (μ) between her shoes and the cliff, we need to consider the maximum force of static friction that can act on the climber. The maximum force of static friction is given by:
Ff(max) = μsN
where μs is the coefficient of static friction. We want to find the minimum value of μs, so we equate Ff(max) to the force the climber must exert with her feet:
Ff(max) = F = 509.6 N
μsN = 509.6 N
Substituting the value of N (509.6 N) into the equation, we have:
μs(509.6 N) = 509.6 N
μs = 1
Therefore, the minimum coefficient of friction (μ) between her shoes and the cliff would be 1.
Note: A coefficient of friction of 1 is the maximum value that static friction can have, indicating a very high frictional force between the climber's shoes and the cliff.
To find the tension in the rope and the force the mountain climber must exert with her feet on the vertical rock face, we can use the concept of equilibrium.
(a) Tension in the rope: In order for the climber to be stationary, the tension in the rope must counteract the force of gravity pulling her downward. The tension in the rope is equal in magnitude to the weight of the climber. The weight of the climber can be calculated using the formula:
Weight = mass x acceleration due to gravity
Given the mass of the climber is 52.0 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight:
Weight = 52.0 kg x 9.8 m/s^2
(b) Force exerted by the feet: To remain stationary, the force exerted by the climber's feet against the rock face must balance out the horizontal component of the tension in the rope. This horizontal component is equal to the tension in the rope multiplied by the sine of the angle between the rope and the vertical direction. As we assume the force is parallel to her legs, this angle is the same as the angle of the rock face.
(c) Minimum coefficient of friction: To calculate the minimum coefficient of friction between her shoes and the cliff, we need to compare the force of friction with the vertical component of the climber's weight. The force of friction can be calculated by multiplying the coefficient of friction by the magnitude of the normal force, which is equal to the vertical component of the weight. The minimum coefficient of friction occurs when the force of friction is at its maximum, which would be equal to the maximum frictional force the shoes can provide.
To find the answer to these questions, we need additional information such as the angle of the rock face and the coefficient of friction between the climber's shoes and the cliff.