Tarzan, who weighs 780 N, swings from a cliff at the end of a 12.0 m vine that hangs from a high tree limb and initially makes an angle of 49.0° with the vertical. Assume that an x axis extends horizontally away from the cliff edge and a y axis extends upward. Immediately after Tarzan steps off the cliff, the tension in the vine is 580 N.

(a) At the moment of takeoff, what is the force on Tarzan from the vine, in unit-vector form?
(b) At that moment, what is the net force on Tarzan, in unit-vector form?

To solve this problem, we can break down the forces into their components. Let's consider the x-axis as horizontal and the y-axis as vertical.

(a) To find the force on Tarzan from the vine at the moment of takeoff, we need to resolve the tension force into its x-component and y-component.

The x-component of the tension force can be found using the sine function:

Fx = Tension * sin(θ)

where θ is the angle the vine makes with the vertical.

Given that Tarzan weighs 780 N and the tension in the vine is 580 N, the x-component of the tension force is:

Fx = Tension * sin(θ)
= 580 N * sin(49.0°)
≈ 370.94 N

The y-component of the tension force can be found using the cosine function:

Fy = Tension * cos(θ)

The y-component of the tension force is:

Fy = Tension * cos(θ)
= 580 N * cos(49.0°)
≈ 424.34 N

Therefore, the force on Tarzan from the vine, in unit-vector form, is:

Fvine = (370.94 N, 424.34 N)

(b) The net force on Tarzan can be obtained by adding the gravitational force and the force from the vine. Since the gravitational force acts vertically downward, its components are:

Fgravity = (0 N, -780 N)

To find the net force, we need to add the x-components and y-components of the forces:

Net Force (Fx, Fy) = (Fx + Fgravity_x, Fy + Fgravity_y)

Net Force (Fx, Fy) = (370.94 N + 0 N, 424.34 N + (-780 N))

Net Force (Fx, Fy) = (370.94 N, -355.66 N)

Therefore, the net force on Tarzan, in unit-vector form, is:

Fnet = (370.94 N, -355.66 N)

To find the force on Tarzan from the vine and the net force on Tarzan, we need to decompose the forces into their x and y components.

(a) The force on Tarzan from the vine can be represented as a vector Fv. Let's break down this force into its x and y components. Since Tarzan swings from the vine, the force Fv can be divided into two components: Fvx (horizontal) and Fvy (vertical).

Fvx = tension in the vine * sin(angle with the vertical)
Fvx = 580 N * sin(49.0°)

Fvy = tension in the vine * cos(angle with the vertical)
Fvy = 580 N * cos(49.0°)

So, the force on Tarzan from the vine, in unit-vector form, is:
Fv = Fvx * î + Fvy * ĵ

(b) The net force on Tarzan can be calculated by considering the gravitational force acting on him. The gravitational force can be represented as a vector Fg. Similar to the force from the vine, we need to decompose this force into x and y components.

Fg = mass of Tarzan * gravitational acceleration
Fg = weight of Tarzan
Fg = 780 N

Fx = 0 (since no horizontal force acts on Tarzan)
Fy = - Fg (negative because it acts downward)

So, the net force on Tarzan, in unit-vector form, is:
Fnet = Fx * î + Fy * ĵ

Note: The î and ĵ are the unit vectors in the x and y directions, respectively. The î unit vector represents the x-axis, and the ĵ unit vector represents the y-axis.

By substituting the values and calculating the magnitudes of both Fv and Fnet, we can find the final answer in unit-vector form.