Question

Jane, whose mass is 50.0kg , needs to swing across a river (having width D ) filled with person-eating crocodiles to save Tarzan from danger. She must swing into a wind exerting constant horizontal force →F , on a vine having length L and initially making an angle θ with the vertical (Fig.P 8.77) . Take D=50.0 m, F = 110 N, L=40.0m , and θ = 50.0°. (b) Once the rescue is complete, Tarzan and Jane must swing back across the river. With what minimum speed must they begin their swing? Assume Tarzan has a mass of 80.0kg .

Answer 1

The minimum speed required for Tarzan and Jane to swing back across the river is approximately 19.80 m/s.

Step-by-step explanation:

To find the minimum speed required for Tarzan and Jane to swing back across the river, we can use the conservation of mechanical energy. If we assume there is no loss of energy due to friction or air resistance, the total mechanical energy at the start of the swing is equal to the total mechanical energy at the end of the swing.

The total mechanical energy is given by:

E = Kinetic Energy + Potential Energy

At the start of the swing, the only form of energy is potential energy: E = mgh, where m is the mass of Tarzan and Jane (130 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the initial height (half the length of the swing, L/2).

At the end of the swing, the only form of energy is kinetic energy: E = (1/2)mv², where v is the velocity at the bottom of the swing.

Setting the two equations for E equal to each other, we can solve for v:

mgh = (1/2)mv²

We can cancel the mass from both sides of the equation:

gh = (1/2)v²

Solving for v:

v = sqrt(2gh)

Substituting the given values, we get:

v = sqrt(2 × 9.8 × 20)

v ≈ 19.80 m/s

Answer 2

Jane needs to start her swing across the river with an initial speed of 0 m/s to successfully save Tarzan. Once the rescue is complete, Tarzan and Jane must begin their swing back across the river with an initial speed of 0 m/s as well.

Step-by-step explanation:

In order to swing across the river, Jane must use the vine to create a pendulum-like motion. The force exerted by the wind can be considered as a horizontal force acting on the vine, which will affect Jane’s swing. We can use the principle of conservation of energy to determine the required speed for Jane to swing across the river.

Let us consider the initial potential energy of the system:

Potential energy (initial) = m g h

where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above the lowest point of the swing.

As the vine is initially inclined at an angle of θ = 50.0°, the height of the system is given by:

h = L * sin(θ)

Substituting this expression for h into the potential energy equation, we get:

Potential energy (initial) = m g L * sin(θ)

Now, let’s consider the final potential energy when Jane reaches the other side of the river:

Potential energy (final) = 0

Since the final height of the system is zero, the final potential energy is also zero.

Now, we can use the principle of conservation of energy to equate the initial and final potential energies:

m g L * sin(θ) = 0

Since the mass (m) and acceleration due to gravity (g) are constants, we can cancel them out:

L * sin(θ) = 0

Since the length (L) of the vine is not zero, we can conclude that:

sin(θ) = 0

This implies that θ = 0°, which means the vine is initially horizontal.

However, this result contradicts the given angle of 50.0°. In this case, we can assume that the wind force is negligible, and we can use the given angle of 50.0°.

Using the given values for L, θ, and the mass of Jane (m = 50.0 kg), we can calculate the initial potential energy:

Potential energy (initial) = m g L sin(θ) Potential energy (initial) = 50.0 kg 9.81 m/s² 40.0 m sin(50.0°) Potential energy (initial) = 13531.2 J

Now, we can use the principle of conservation of energy to find the final kinetic energy of the system:

Kinetic energy (final) = Potential energy (initial) Kinetic energy (final) = 13531.2 J

Since the final kinetic energy is equal to the final potential energy, which is zero, the system must come to a complete stop. Therefore, Jane needs to start her swing with a speed of:

Initial speed = √(2 Kinetic energy (final)) Initial speed = √(2 0) Initial speed = 0 m/s

Jane needs to start her swing with an initial speed of 0 m/s to successfully cross the river.

Now, let’s find the minimum speed with which Tarzan and Jane must begin their swing back across the river. We can use the same principle of conservation of energy as before, but this time, we must consider the combined mass of Tarzan and Jane (m = 80.0 kg).

Using the same method as in section 2, we can find the minimum speed with which they must start their swing:

Initial speed = √(2 Kinetic energy (final)) Initial speed = √(2 0) Initial speed = 0 m/s

Tarzan and Jane must begin their swing back across the river with an initial speed of 0 m/s.