Tarzan tries to cross a river by swinging from one bank to the other on a vine that is 9.0 m long. His speed at the bottom of the swing is 8.8 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?
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To determine the largest mass that Tarzan can have and still make it safely across the river, we need to calculate the tension in the vine when it is at its maximum stress.
First, we can find the height at the top of Tarzan's swing using the conservation of mechanical energy. At the bottom of the swing, Tarzan has maximum kinetic energy, which is converted to maximum potential energy at the top of the swing.
Using the equation: mgh = (1/2)mv^2
Where:
m = mass
g = acceleration due to gravity (9.8 m/s^2)
h = height
At the bottom of the swing, Tarzan's speed is given as 8.8 m/s. We can rearrange the equation and solve for h:
h = (v^2)/(2g)
h = (8.8^2) / (2 * 9.8)
h ≈ 3.94 m
Now, we can calculate the maximum tension in the vine. At the top of the swing, Tarzan is momentarily weightless, and the only forces acting on him are the tension in the vine T and his weight mg.
Using the equation: T = mg
The tension T can be calculated by multiplying Tarzan's mass by the acceleration due to gravity g:
T = mg = 1.0 * 10^3 kg * 9.8 m/s^2
T ≈ 9.8 * 10^3 N
Therefore, the largest mass that Tarzan can have and still make it safely across the river is equal to the breaking strength of the vine, which is 1.0 * 10^3 N.