consider the 52 kg climber. find the tension in the rope and the force that the 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. what is the minimum coefficient of friction between her shoes and cliff?
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To find the tension in the rope and the force the climber must exert with her feet on the vertical rock face, we can use the concept of equilibrium.
First, let's consider the forces acting on the climber:
1. Weight (W) acting vertically downward = mass (m) * gravity (g)
W = 52 kg * 9.8 m/s^2 = 509.6 N
Note: This force acts downward and is balanced by the tension in the rope.
2. Tension in the rope (T) acting vertically upward
The tension in the rope is equal in magnitude and opposite in direction to the weight of the climber.
T = 509.6 N
3. Force exerted by the climber's feet (F) acting perpendicular to the vertical rock face
This force is directed upwards and balances a component of gravity.
4. Force of friction (Ff) acting parallel to the vertical rock face
This force opposes the motion of the climber.
Now, let's find the force exerted by the climber's feet:
Since the climber is stationary, the vertical forces must be balanced:
F + T = W
F + 509.6 N = 509.6 N
Hence, the force exerted by the climber's feet is:
F = 0 N
Since the climber is not exerting any force with her feet, we can conclude that the minimum coefficient of friction (μ) required between her shoes and the cliff face is 0.
Note: This result implies that the climber's shoes must have a coefficient of friction greater than or equal to zero to remain stationary. The climbing shoes should provide enough friction to counteract the force that would cause the climber to slip downwards.
To find the tension in the rope and the force that the climber must exert with her feet on the vertical rock face to remain stationary, we can analyze the forces acting on the climber.
1. Tension in the Rope:
The tension in the rope is equal to the weight of the climber. Since the weight is given as 52 kg, we can calculate the tension using the formula:
Tension = Weight = mass × gravity
Here, the acceleration due to gravity, g, is approximately 9.8 m/s². So, the tension in the rope is:
Tension = 52 kg × 9.8 m/s² = 509.6 N
Therefore, the tension in the rope is 509.6 N.
2. Force Exerted with Feet:
To find the force the climber exerts with her feet on the rock face, we first need to consider the forces acting on her. The two forces acting vertically are the tension in the rope (upward force) and the weight of the climber (downward force).
Since the climber is stationary, the net force in the vertical direction is zero. Therefore, the force exerted with her feet on the rock face must be equal in magnitude and opposite in direction to the tension in the rope.
Thus, the force exerted with her feet on the rock face is also 509.6 N.
3. Minimum Coefficient of Friction:
The force exerted with her feet on the rock face is equal to the product of the coefficient of friction (μ) and the normal force (N). The normal force, in this case, is equal to the weight of the climber since the vertical forces are balanced.
Since we know the force exerted with her feet is 509.6 N and the weight of the climber is also 509.6 N, we can set up the equation:
Force exerted = Coefficient of friction × Normal force
509.6 N = μ × 509.6 N
The weight of the climber cancels out, and we find:
1 = μ
Therefore, the minimum coefficient of friction between her shoes and the cliff is 1.