1. First, we need to identify the forces acting on the mountain climber. There are three forces: gravitational force (mg), tension force (T) in the rope, and the normal force (N) exerted by the slope on the climber.
2. Next, we need to break down these forces into components along the slope and perpendicular to the slope. The gravitational force has two components: mg sin(angle) along the slope and mg cos(angle) perpendicular to the slope. The tension force is along the slope, and the normal force is perpendicular to the slope.
3. Now, we can apply Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration. Since the climber is not moving along the slope, the net force along the slope is zero. Therefore, we can write the equation: T - mg sin(angle) = 0
4. Solving for the tension force, we get: T = mg sin(angle) This is the formula for the tension in the rope when the slope is inclined at an angle above the horizontal.
5. Now, let's check our results for the limits angle = 0 degrees and angle = 90 degrees. a) When angle = 0 degrees, sin(0) = 0, so the tension force is: T = mg * 0 = 0 This makes sense because when the slope is horizontal, there is no need for the climber to hold onto the rope to prevent sliding. b) When angle = 90 degrees, sin(90) = 1, so the tension force is: T = mg * 1 = mg In this case, the climber is hanging vertically from the rope, and the tension force must be equal to the gravitational force to keep the climber from falling.
