Tarzan (m = 78.4 kg) tries to cross a river by swinging from a vine. The vine is 10 m
long. Tarzan doesn't know that the vine has a breaking strength of 1045 N. What
maximum speed (in m/s) can Tarzan have at the bottom of the swing (as he just clears
the water) in order to safely cross the river without breaking the vine?
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To find the maximum speed at the bottom of the swing that Tarzan can have without breaking the vine, we need to analyze the forces acting on Tarzan during his swing.
1. First, let's consider the forces at the bottom of the swing:
- Gravity (mg): This force acts vertically downward and has a magnitude equal to the product of Tarzan's mass (m) and the acceleration due to gravity (g ≈ 9.8 m/s^2).
- Tension in the vine (T): This force acts along the length of the vine and provides the centripetal force required to keep Tarzan in circular motion.
2. At the bottom of the swing, the net force acting on Tarzan is the difference between the tension in the vine and the gravitational force:
Net force (F_net) = T - mg
3. To keep Tarzan safe, the net force must be less than or equal to zero since any excess force will cause the vine to break. Therefore, we have:
F_net ≤ 0
T - mg ≤ 0
T ≤ mg
4. The maximum tension in the vine occurs when Tarzan is at the bottom of the swing. Thus, the maximum tension that the vine can withstand is 1045 N.
5. Substituting the values, we have:
1045 N ≤ m * g
1045 N ≤ 78.4 kg * 9.8 m/s^2
6. Solving for m * g, we get:
m * g = 768.32 N
7. Now, we need to find the maximum speed at the bottom of the swing (v_max). We know that the centripetal force required for circular motion is given by:
F_c = m * v^2 / r
In this case, the radius (r) is equal to the length of the vine, which is 10 m. We also know that the centripetal force is provided by the tension (T) in the vine. Therefore, we can write:
T = m * v^2 / r
Rearranging the equation, we have:
v^2 = T * r / m
8. Substituting the known values, we have:
v^2 = (1045 N) * (10 m) / (78.4 kg)
v^2 = 1327.806 m^2/s^2
9. Finally, taking the square root of both sides, we find:
v ≈ 36.45 m/s
Therefore, the maximum speed Tarzan can have at the bottom of the swing (just clearing the water) to safely cross the river without breaking the vine is approximately 36.45 m/s.