We can first find the x and y components of the tension force in the vine.
(a) T_x = T*cos(22°) = 760*cos(22°) and T_y = T*sin(22°) = 760*sin(22°).
Calculating these values, we get T_x = 719.4 N and T_y = 287.69 N. Thus, the force on Tarzan from the vine in unit-vector notation is F_T = 719.4 i + 287.69 j (N).
(b) Now we need to find the net force on Tarzan. Since Tarzan weighs 820 N, there is a gravitational force F_g = -820 j (N) acting on him. The net force is the vector sum of the tension force and the gravitational force: F_net = F_T + F_g.
F_net = (719.4 i + 287.69 j) + (0 i - 820 j) = 719.4 i - 532.31 j (N)
(c) To find the magnitude of the net force, we use the Pythagorean theorem: |F_net| = sqrt((719.4)^2 + (-532.31)^2) = 901.17 N
(d) To find the angle of the net force relative to the positive direction of the x-axis, we use the tangent function: tan(theta) = (-532.31) / (719.4).
Solving for theta, we find theta = atan(-532.31 / 719.4) = -36.5°.
(e) Tarzan's acceleration can be found using Newton's second law: F_net = m*a, where m is Tarzan's mass and a is the acceleration vector.
First, we need to find Tarzan's mass. Weight = mass * g (gravity), so mass = Weight / g = 820 N / 9.81 m/s^2 = 83.6 kg.
Now we can find Tarzan's acceleration components in x and y: a_x = F_net_x / m = 719.4 N / 83.6 kg = 8.6 m/s^2, and a_y = F_net_y / m = -532.31 N / 83.6 kg = -6.36 m/s^2.
The magnitude of Tarzan's acceleration is |a| = sqrt((8.6 m/s^2)^2 + (-6.36 m/s^2)^2) = 10.68 m/s^2.
(f) The angle of Tarzan's acceleration relative to the positive direction of the x-axis can be found using the tangent function again: tan(theta) = -6.36 / 8.6.
Solving for theta, we find theta = atan(-6.36 / 8.6) = -36.5° (the same as the angle of the net force).
Tarzan, who weighs 820 N, swings from a cliff at the end of a 20.0 m vine that hangs from a high tree limb and initially makes an angle of 22.0° 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 760 N. Just then, what are (a) the force on him from the vine in unit-vector notation and the net force on him (b) in unit-vector notation and as (c) a magnitude and (d) an angle relative to the positive direction of the x axis? What are the (e) magnitude and (f) angle of Tarzan’s acceleration just then?
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