“Bob traverses a chasm by stringing a rope between a tree on one side of the chasm and a tree on the opposite side, 25 m away. Assume the rope can provide a tension force of up to 29 kN before breaking”
The Correct Answer and Explanation is :
To determine if Bob can safely traverse the chasm by stringing the rope between the two trees, we need to analyze the forces involved, specifically focusing on the tension in the rope. We are given the following:
- The distance between the trees is 25 meters.
- The rope can handle a maximum tension of 29 kN (kilonewtons).
Step 1: Identify Key Factors
Bob will likely create an angle when he crosses the rope (it won’t be completely horizontal). This angle will affect the tension the rope experiences because the tension force in a rope depends on both the angle of the rope and the vertical force due to Bob’s weight.
Step 2: Resolve Forces in the Rope
If we consider the tension in the rope, we have to account for two components of the tension force:
- The vertical component, which balances Bob’s weight.
- The horizontal component, which keeps the rope taut between the trees.
The tension force ( T ) can be broken into its vertical and horizontal components:
- Vertical component: ( T_{\text{vertical}} = T \sin(\theta) )
- Horizontal component: ( T_{\text{horizontal}} = T \cos(\theta) )
Where ( \theta ) is the angle the rope makes with the horizontal.
Step 3: Considering Bob’s Weight
Let’s assume Bob weighs 700 N (a reasonable average weight). The vertical component of the tension must balance his weight to prevent the rope from breaking or Bob from falling. Therefore:
[ T_{\text{vertical}} = 700 \, \text{N} ]
We can set up the equation:
[ T \sin(\theta) = 700 \, \text{N} ]
Step 4: Checking the Tension Limit
Now, we know the maximum tension the rope can withstand is 29 kN (29,000 N). The actual tension required to support Bob’s weight will depend on the angle ( \theta ), but the key point is whether the rope can handle the required tension.
Since the maximum tension is 29,000 N and the vertical component needs to balance only 700 N, the rope can easily support Bob’s weight even with some angle between the trees. However, if Bob were to add additional forces (like swinging), the tension could exceed this limit, leading to the risk of breaking the rope.
Conclusion
The rope will likely be able to support Bob’s weight given the maximum tension of 29 kN. However, it’s essential that Bob keeps the rope as close to horizontal as possible to minimize the angle and reduce the vertical tension needed. If the angle becomes too steep or Bob exerts additional force (such as jumping), the rope might reach its breaking point. Hence, Bob must ensure he is cautious to avoid exceeding the rope’s tension limit.