The tension in a rope attached to a boat is 27.5 lb. The
rope is attached to the boat 6.0 ft below the level at
which the boat is being drawn in. At the point where
there is 18.0 ft of rope out, what force is bringing the
boat towards the dock, and what force tends to raise
the boat? What if there is 9 ft of rope out?
To determine the force bringing the boat towards the dock and the force that tends to raise the boat, we need to break down the tension force in the rope into its vertical and horizontal components.
Let's start by determining the force bringing the boat towards the dock when there is 18.0 ft of rope out:
1. Vertical Component: The force that tends to raise the boat can be found by calculating the vertical component of the tension force. Since the rope is attached 6.0 ft below the boat, the vertical distance is 6.0 ft. We can use the equation: Vertical Component = Tension * (vertical distance / rope length).
Vertical Component = 27.5 lb * (6.0 ft / 18.0 ft) = 9.17 lb.
2. Horizontal Component: The horizontal component of the tension force is responsible for bringing the boat towards the dock. It is equal to the tension force itself since there is no horizontal displacement of the rope.
Horizontal Component = Tension = 27.5 lb.
Therefore, when there is 18.0 ft of rope out, the force bringing the boat towards the dock is 27.5 lb, and the force that tends to raise the boat is 9.17 lb.
Now let's calculate the forces when there is only 9 ft of rope out:
1. Vertical Component: Using the same equation as before, the vertical component can be calculated as follows:
Vertical Component = 27.5 lb * (6.0 ft / 9.0 ft) = 18.33 lb.
2. Horizontal Component: Since the rope length is now shorter, the horizontal component will also decrease proportionally:
Horizontal Component = Tension * (horizontal distance / rope length).
Horizontal Component = 27.5 lb * (9.0 ft / 9.0 ft) = 27.5 lb.
Therefore, when there is 9 ft of rope out, the force bringing the boat towards the dock is 27.5 lb, and the force that tends to raise the boat is 18.33 lb.