Tarzan spies a 35kg chimpanzee in severe danger, so he swings to the rescue. He adjusts his strong, but very light, vine so that he will first come to rest 4.0 s after beginning his swing, at which time his vine makes a 12 degree angle with the vertical. (a) How long is Tarzan's vine, assuming that he swings at the bottom end of it? (b) What are the frequency and amplitude (in degrees) of Tarzan's swing? (c) Just as he passes through the lowest point in his swing, Tarzan nabs the chimp from the ground and sweeps him out of the jaws of danger. If Tarzan's mass is 65 kg, find the frequency and amplitude (in degrees) of the swing with Tarzan holding onto the grateful chimp.
To solve this problem, we can use the principle of conservation of mechanical energy and the equation of motion for simple harmonic motion.
(a) To find the length of Tarzan's vine, we can use the equation for the period of a simple pendulum:
T = 2π√(L/g),
where T is the time period, L is the length of the pendulum, and g is the acceleration due to gravity.
Given that Tarzan comes to rest 4.0 s after beginning his swing, we can determine the time period:
T = 4.0 s.
We are also given that the vine makes a 12-degree angle with the vertical when Tarzan comes to rest. To find the length of the vine, we can use trigonometry:
sin(12°) = (L/g) / T.
Rearranging the equation, we get:
L = (g * T^2) / (4π^2 * sin(12°)).
Substituting the known values of g = 9.8 m/s^2 and T = 4.0 s, we can calculate the length:
L = (9.8 * (4.0^2)) / (4π^2 * sin(12°)).
(b) To find the frequency and amplitude of Tarzan's swing, we need to use the formulas for frequency and amplitude of simple harmonic motion:
Frequency (f) = 1 / T,
Amplitude (A) = θ_max,
where T is the time period and θ_max is the maximum angle of displacement.
Given that T = 4.0 s, we can calculate the frequency:
f = 1 / 4.0 = 0.25 Hz.
The maximum angle of displacement (θ_max) is given as 12 degrees.
Therefore, the frequency of Tarzan's swing is 0.25 Hz, and the amplitude is 12 degrees.
(c) When Tarzan grabs the chimp, the system becomes a combination of a simple pendulum (Tarzan's body) and a mass (chimpanzee) attached to it. The overall frequency and amplitude of the swing will change due to the addition of the chimpanzee's mass.
To find the new frequency and amplitude, we can use the equation for the period of a combined pendulum-mass system:
T = 2π√(I/mg),
where T is the time period, I is the moment of inertia of Tarzan and the chimp together about the swing axis, m is the total mass, and g is the acceleration due to gravity.
Given that Tarzan's mass is 65 kg and the chimp's mass is 35 kg, the total mass of the system is:
m = 65 kg + 35 kg = 100 kg.
We can assume that Tarzan's body behaves like a simple pendulum, so the moment of inertia about the swing axis is:
I = ml^2,
where l is the distance from the swing axis to Tarzan's center of mass.
Substituting the known values, we can calculate the moment of inertia:
I = 65 kg * (L/2)^2.
Now, we can determine the new time period:
T = 2π√(I/mg).
Finally, we can use the time period to find the frequency and the maximum angle of displacement:
Frequency (f) = 1 / T,
Amplitude (A) = θ_max.
Therefore, the frequency and amplitude of the swing with Tarzan holding onto the chimp can be calculated using the above equations.
To solve this problem, we can use the principles of circular motion and the equations of motion. Let's go through each part of the problem step-by-step:
(a) How long is Tarzan's vine, assuming that he swings at the bottom end of it?
To find the length of the vine, we can start by analyzing the forces acting on Tarzan at the bottom end of his swing. At the lowest point, the tension in the vine provides the centripetal force required for circular motion. The weight of Tarzan acts downward, and the tension force acts upward.
Since Tarzan is at rest at the topmost point of his swing, we have two equilibrium equations:
1. Equating the vertical forces:
T - mg = 0 -----> Equation 1
2. Equating the net force required for centripetal motion:
T = mv^2 / r -----> Equation 2
Where:
T = Tension in the vine
m = Mass of Tarzan
g = Acceleration due to gravity
v = Velocity of Tarzan at the lowest point (which is zero)
r = Length of the vine (to be determined)
From Equation 1, we can solve for the tension T:
T = mg -----> Equation 3
Substituting Equation 3 into Equation 2:
mg = (m * 0^2) / r
mg = 0 / r
r = ∞
Since r turns out to be infinite, this means that Tarzan's vine has no fixed length. However, this is not physically possible. Therefore, we must assume that Tarzan does not swing from the bottom end of the vine.
(b) What are the frequency and amplitude (in degrees) of Tarzan's swing?
Since Tarzan doesn't swing from the bottom end of the vine, we cannot determine the frequency and amplitude of his swing without knowing the actual length of the vine and the angle he starts swinging from.
(c) Just as he passes through the lowest point in his swing, Tarzan nabs the chimp from the ground and sweeps him out of the jaws of danger. If Tarzan's mass is 65 kg, find the frequency and amplitude (in degrees) of the swing with Tarzan holding onto the grateful chimp.
To find the frequency and amplitude of Tarzan's swing with the chimp, we need to consider the new system's total mass, which includes Tarzan and the chimp. Let's assume the mass of the chimp is m_chimp.
The total mass, m_total, is given by:
m_total = m_tarzan + m_chimp
Given:
m_tarzan = 65 kg
m_chimp = 35 kg
m_total = 65 kg + 35 kg = 100 kg
At the lowest point of Tarzan's swing, the tension in the vine provides the necessary centripetal force for circular motion. The weight of the combined system acts downward, and the tension force acts upward.
Equating the vertical forces:
T - m_total * g = 0 -----> Equation 4
Now, we can solve Equation 4 for the tension T:
T = m_total * g -----> Equation 5
At the lowest point, Tarzan's velocity is zero again, and we can use Equation 2 to find the total length of the vine, r.
T = m_total * v^2 / r -----> Equation 6
Now, we can substitute T from Equation 5 into Equation 6:
m_total * g = m_total * 0^2 / r
m_total * g = 0 / r
r = ∞
Similar to part (a), we find that r is infinite, which means the vine has no fixed length. However, this is not physically possible.
Therefore, we need additional information, such as the angle at which Tarzan swings or the length of the vine, to calculate the frequency and amplitude of Tarzan's swing with the chimp.
In conclusion, without further information, we cannot determine the frequency and amplitude of Tarzan's swing with the chimp.
(a) For the length L of the vine, use the fact that the period of oscillation (back and forth) is
P = 8.0 s = 2 pi sqrt(L/g)
g is the acceleration of gravity. Solve gor L
b) Picking up the chimp will not affect the frequency/period of vibration, and it will be an inelastic collision process with momentum conserved. The two of them will have their speed at the bottom of the swing reduced by a ratio 65/100. The kinetic energy will be reduced by a (100/65)(65/100)^2 = 0.65 factor
This will reduce the swing amplitude so that
1 - cos theta) is 65% of what it was before, since maximum potential energy is proportional to (1 - cos theta)
Original 1 - cos theta = 0.0219
Final 1 - cos theta = 0.0142
cos theta = 0.9858
theta = 9.7 degrees.