Tarzan tries to cross a river by swinging from one bank to the other on a vine that is 8.2 m long. His speed at the bottom of the swing is 7.2 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?
tension=mg+mv^2/r
set that equal to breaking strength, and solve for mass m
To determine 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 his swing.
1. First, let's calculate the tension in the vine when Tarzan is at the bottom of his swing. At the bottom of the swing, Tarzan's centripetal acceleration is given by:
a = v^2 / r
where v is his speed (7.2 m/s) and r is the length of the vine (8.2 m).
Substituting the values, we get:
a = (7.2 m/s)^2 / 8.2 m = 6.28 m/s^2 (approximately)
2. The force due to centripetal acceleration is given by the equation:
F = ma
where m is the mass of Tarzan and a is the centripetal acceleration.
Rearranging the formula, we get:
m = F / a
3. Now, let's calculate the centripetal force acting on Tarzan:
F = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting for F, we get:
m * g = F
4. The tension force in the vine acts as the centripetal force keeping Tarzan moving in a circular path. Therefore, the tension force must be equal to or smaller than the breaking strength of the vine:
T <= 1.0 * 10^3 N
5. Combining equations from step 2, 3, and 4, we have:
m * g = T <= 1.0 * 10^3 N
Now, let's solve for the maximum mass, m:
m = T / g
m <= 1.0 * 10^3 N / 9.8 m/s^2
m <= 102 kg (approximately)
Therefore, the largest mass that Tarzan can have and still make it safely across the river is approximately 102 kg.