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?



kg

Fcentripetal = mAc = m v^2/R

= m (7.2)^2/8.2
Fgravity = m g = m (9.81)
so total force on vine =

m ( 9.81 + 7.2^2/8.2) = 10^3 Newtons

solve for m

Note: Tarzan just swam across the wild river from rock to rock with a vine around his waist and the natives are rebuilding their bridge now as we speak.

see:

http://www.gocomics.com/tarzan?ref=comics

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 Tarzan during the swing.

When Tarzan is at the bottom of the swing, the tension in the vine must be equal to his weight plus the centripetal force required to keep him moving in a circular path.

Let's calculate the tension in the vine at the bottom of the swing:

1. First, we calculate the centripetal force using the formula:

F = m * a

Where F is the centripetal force, m is the mass, and a is the centripetal acceleration.

The centripetal force can be calculated as the product of the mass and the square of the velocity, divided by the radius of the swing path:

F = (m * v^2) / r

We know the velocity at the bottom of the swing (v = 7.2 m/s) and the radius of the swing path (r = 8.2 m). So, we can substitute these values into the formula:

F = (m * (7.2 m/s)^2) / 8.2 m

2. Next, we add Tarzan's weight to the centripetal force to calculate the tension in the vine:

T = F + mg

Where T is the tension in the vine, m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2).

So, the equation becomes:

T = (m * (7.2 m/s)^2) / 8.2 m + m * 9.8 m/s^2

3. Finally, we set the tension in the vine equal to the breaking strength of the vine:

T = 1.0 × 10^3 N

Now, we can solve the equation for m, the largest mass Tarzan can have:

(m * (7.2 m/s)^2) / 8.2 m + m * 9.8 m/s^2 = 1.0 × 10^3 N

Simplifying the equation:

(m * 51.84 m^2/s^2) / 8.2 m + 9.8 m/s^2 * m = 1.0 × 10^3 N

Cross-multiplying and rearranging the equation:

51.84 m^2/s^2 + 9.8 m/s^2 * 8.2 m = 1.0 × 10^3 N

51.84 m^2/s^2 + 80.36 m^2/s^2 = 1.0 × 10^3 N

132.2 m^2/s^2 = 1.0 × 10^3 N

m = (1.0 × 10^3 N) / (132.2 m^2/s^2)

m ≈ 7.56 kg

Therefore, the largest mass that Tarzan can have and still make it safely across the river is approximately 7.56 kg.