Tarzan tries to cross a river by swinging from one bank to the other on a vine that is 11.2 m long. His speed at the bottom of the swing is 7.0 m/s. Tarzan does not know that the vine has a breaking strength of 1.0 ✕ 103 N. What is the largest mass that Tarzan can have and still make it safely across the river?
To determine the largest mass that Tarzan can have and still make it safely across the river, we need to consider the physics involved in this scenario.
Given:
- Length of the vine (L) = 11.2 m
- Speed at the bottom of the swing (v) = 7.0 m/s
- Breaking strength of the vine (F) = 1.0 × 10^3 N
Let's break down the problem into steps:
Step 1: Calculate the maximum tension in the vine.
When Tarzan is swinging, the tension in the vine is maximum at the bottom of the swing. At this point, the tension force (T) in the vine consists of two components:
- The weight of Tarzan (mg) acting downwards.
- The centripetal force (mv^2 / L) acting towards the center of the swing.
So, we have:
T = mg + mv^2 / L
Step 2: Equate the tension to the breaking strength of the vine.
To ensure that Tarzan can cross the river safely, the tension in the vine (T) should not exceed its breaking strength (F).
Therefore, we set:
T ≤ F
Step 3: Substitute the values and solve for mass (m).
Using the equations from steps 1 and 2, we can substitute the given values to find the maximum mass (m) that Tarzan can have.
mg + mv^2 / L ≤ F
Substituting the given values:
m * 9.8 + m * (7.0^2) / 11.2 ≤ 1.0 × 10^3
Simplifying the equation:
9.8m + 49m / 11.2 ≤ 1.0 × 10^3
Combining like terms:
(9.8 * 11.2 + 49)m / 11.2 ≤ 1.0 × 10^3
m ≤ (109.76 / 11.2) ≤ 98.0 kg
Therefore, the largest mass that Tarzan can have and still make it safely across the river is 98.0 kg.