A 75 kg rock climber is attached to a rope that is allowing her to hang horizontally with her feet against the wall. The tension in the rope is 825 N and the rope makes a 35 degree angle with the vertical. Determine the force exerted by the wall on the climbers feet.
Force exerted by the wall on the climber's feet = (825 N * cos 35°) = 717.5 N
To determine the force exerted by the wall on the climber's feet, we need to analyze the forces acting on the climber in equilibrium.
Let's break down the forces acting on the climber:
1. Weight: The weight of the climber acts vertically downward, and its magnitude is given by the formula W = m * g, where m is the mass (75 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2). Therefore, the weight of the climber is W = 75 kg * 9.8 m/s^2 = 735 N.
2. Rope Tension: The tension in the rope acts horizontally and is directed toward the wall. Its magnitude is given as 825 N.
3. Force exerted by the wall: This force acts vertically and is exerted by the wall on the climber's feet. We need to find this force.
Now, let's analyze the vertical and horizontal forces:
Vertical Forces:
The vertical component of the rope tension will balance out the weight of the climber. Since the climber is hanging horizontally against the wall, the vertical component of the rope tension provides the upward force.
Vertical Force: Tension_vertical = Tension * sin(theta)
Vertical Force: Tension_vertical = 825 N * sin(35 degrees)
Horizontal Forces:
The horizontal component of the rope tension provides the horizontal force and equals the force exerted by the wall on the climber.
Horizontal Force: Tension_horizontal = Tension * cos(theta)
Horizontal Force: Tension_horizontal = 825 N * cos(35 degrees)
To find the force exerted by the wall on the climber's feet, we only need to consider the vertical component because the horizontal component doesn't affect it.
Therefore, the force exerted by the wall on the climber's feet is approximately:
Force_exerted_by_wall = Tension * sin(theta)
Force_exerted_by_wall = 825 N * sin(35 degrees) ≈ 470.02 N
So, the force exerted by the wall on the climber's feet is approximately 470.02 Newtons.
To determine the force exerted by the wall on the climber's feet, we can use the concept of force components.
Step 1: Draw a free-body diagram for the climber, considering the forces acting on her.
Tension (T)
/
/
/ Climber (75 kg)
/
/
Wall (Force exerted by the wall on the climber's feet)
|
|
Gravity (Weight of the climber, mg)
Step 2: Resolve the tension force (T) into vertical and horizontal components. The vertical component of the tension force is responsible for cancelling out the gravitational force acting on the climber, while the horizontal component of the tension force is responsible for exerting the force against the wall.
The vertical component of the tension force (T_v) can be calculated using trigonometry:
T_v = T * cos(35°)
The horizontal component of the tension force (T_h) can also be calculated using trigonometry:
T_h = T * sin(35°)
Step 3: Calculate the weight of the climber, which is the force due to gravity acting vertically downward:
Weight (W) = mass (m) * gravity (g)
= 75 kg * 9.8 m/s²
Step 4: Equate the vertical component of the tension force (T_v) with the weight of the climber (W):
T_v = W
Step 5: Calculate the force exerted by the wall on the climber's feet (Wall):
Wall = T_h
Substituting the values into the equations:
T_v = T * cos(35°)
= 825 N * cos(35°)
T_h = T * sin(35°)
= 825 N * sin(35°)
W = 75 kg * 9.8 m/s²
Wall = T_h
After performing the calculations, the force exerted by the wall on the climber's feet can be determined.