Questions: An adventurous archaeologist crosses between two rock cliffs by slowly going hand over hand along a rope stretched between the cliffs. He stops to rest at the middle of the rope. The rope will break if the tension in it exceeds 2.50 x 10^4 N. Our hero's mass is 70.0 kg.
What is the smallest value the angle θ can have if the rope is not to break?
Express your answer in degrees.
Transcript text: An adventurous archaeologist crosses between two rock cliffs by slowly going hand over hand along a rope stretched between the cliffs. He stops to rest at the middle of the rope. The rope will break if the tension in it exceeds $2.50 \times 10^{4} \mathrm{~N}$. Our hero's mass is 70.0 kg.
What is the smallest value the angle $\theta$ can have if the rope is not to break?
Express your answer in degrees.
Solution
Solution Steps
Step 1: Identify the Given Data
Mass of the hero, m=70.0kg
Maximum tension the rope can withstand, Tmax=2.50×104N
Gravitational acceleration, g=9.8m/s2
Step 2: Calculate the Weight of the Hero
The weight W of the hero is given by:
W=m⋅gW=70.0kg×9.8m/s2W=686N
Step 3: Analyze the Forces in the Rope
At the midpoint, the tension in the rope has two components due to symmetry. Each side of the rope supports half of the weight of the hero. The vertical component of the tension T must balance the weight of the hero:
2Tsin(θ)=WTsin(θ)=2WTsin(θ)=2686NTsin(θ)=343N
Step 4: Determine the Minimum Angle θ
The tension T must not exceed the maximum tension Tmax:
T≤TmaxT=sin(θ)343Nsin(θ)343N≤2.50×104Nsin(θ)≥2.50×104N343Nsin(θ)≥0.01372
Step 5: Calculate the Minimum Angle θ
θmin=sin−1(0.01372)θmin≈0.785∘
Final Answer
The smallest value the angle θ can have if the rope is not to break is approximately 0.785∘.