Tarzan is crossing a river by swinging on a 10.0-m long vine with a breaking point of 1.0 x 10^3 N. If Tarzan's speed at the bottom of his swing is 8.0 m/s, what is the largest mass that Tarzan can have and still make it across the river? [Hint: There are multiple forces acting on Tarzan.]

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To solve this problem, we need to consider the forces acting on Tarzan as he swings across the river. The primary forces involved are the tension in the vine and the gravitational force.

The tension in the vine is the force that Tarzan exerts on the vine to keep it taut while swinging. The gravitational force is the force pulling Tarzan downwards.

At the bottom of Tarzan's swing, the tension in the vine reaches its maximum value because both forces are acting in the same direction. To determine the maximum mass Tarzan can have, we need to calculate the tension in the vine at this point.

We can start by calculating the tension using the formula for gravitational force:

F_gravity = m * g

where F_gravity is the gravitational force, m is Tarzan's mass, and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Rearranging the formula, we get:

m = F_gravity / g

Next, we can determine the maximum tension in the vine at the bottom of Tarzan's swing. Since the vine has a breaking point, the maximum tension must not exceed the breaking point.

To find the tension in the vine, we can use the centripetal force equation:

F_centr = m * v^2 / r

where F_centr is the centripetal force, m is Tarzan's mass, v is Tarzan's speed at the bottom of the swing, and r is the length of the vine.

Rearranging the formula, we get:

m = F_centr * r / v^2

We now have two equations for m, the mass of Tarzan. We can equate the two expressions to determine the maximum value of m that allows Tarzan to make it across the river:

F_gravity / g = F_centr * r / v^2

Substituting the given values (F_gravity = m * g = m * 9.8 m/s^2, r = 10.0 m, and v = 8.0 m/s), we can solve for m:

m * 9.8 m/s^2 / g = m * 10.0 m / (8.0 m/s)^2

Simplifying, we find:

9.8 / g = 10.0 / 64

Solving for g, we have:

g = 64 * 9.8 / 10.0

g ≈ 62.72 m/s^2

Now we can substitute the value of g into the equation to find m:

9.8 / 62.72 = m * 10.0 / (8.0)^2

Simplifying further, we find:

m ≈ (9.8 * 64) / (620.8)

m ≈ 1.02 kg

Therefore, the largest mass that Tarzan can have and still make it across the river is approximately 1.02 kg.