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?
To find the largest mass that Tarzan can have and still make it safely across the river, we need to consider the forces acting on him during the swing.
1. Determine the tension in the vine:
Tarzan is swinging on a vine, so the tension in the vine will be providing the centripetal force necessary for the circular motion. We can use the centripetal force equation:
Centripetal Force = (mass × velocity^2) / radius
The radius of the circular motion is the length of the vine, given as 9.0 m.
We need to find the tension in the vine at the bottom of the swing when the velocity is 8.8 m/s.
Substituting the known values:
Tension = (mass × velocity^2) / radius
2. Ensure that the tension is less than the breaking strength of the vine:
The tension in the vine must be less than or equal to the breaking strength, which is given as 1.0 × 10^3 N.
3. Solve for the maximum mass:
Now, rearranging the formula for tension to solve for mass:
Mass = (tension × radius) / velocity^2
Substituting the given values:
Mass = (1.0 × 10^3 N × 9.0 m) / (8.8 m/s)^2
Calculating the value will give us the largest mass that Tarzan can have and still make it safely across the river.