Tarzan swings on a 27.0 m long vine initially inclined at an angle of 35.0° with the vertical.(a) What is his speed at the bottom of the swing if he starts from rest?
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To find Tarzan's speed at the bottom of the swing, we can use the conservation of mechanical energy.
The total mechanical energy at the top of the swing is equal to the total mechanical energy at the bottom of the swing.
At the top of the swing, Tarzan's energy is purely potential energy:
Potential Energy (PE) = m * g * h
At the bottom of the swing, Tarzan's energy is a combination of kinetic energy and potential energy:
Total Mechanical Energy (E) = KE + PE
Assuming Tarzan starts from rest, the initial kinetic energy is zero.
Therefore, we can write the equation as:
Potential Energy at the top = Total Mechanical Energy at the bottom
m * g * h = KE + PE
Given data:
Length of the vine (h) = 27.0 m
Angle with the vertical (θ) = 35.0°
To calculate the height (h), we use trigonometry:
h = length of the vine * sin(θ)
h = 27.0 m * sin(35.0°)
Let's calculate h:
h = 27.0 m * sin(35.0°)
h ≈ 15.61 m
Substituting this value into the equation:
m * g * h = KE + PE
m * g * 15.61 m = 0 + PE
We know that m * g is the gravitational potential energy (PE):
PE = m * g * h
Now, we can rearrange the equation to solve for speed (v) at the bottom of the swing:
KE = PE - m * g * h
(1/2) * m * v^2 = m * g * h - m * g * h
(1/2) * m * v^2 = 0
Since v is squared in the equation, we can conclude that the speed at the bottom of the swing is zero. This means Tarzan comes to a momentary stop at the bottom before swinging back up.
To find Tarzan's speed at the bottom of the swing, we can use the conservation of mechanical energy.
1. The first step is to calculate the potential energy at the top of the swing. Since Tarzan starts from rest, all of his initial mechanical energy is in the form of potential energy. The formula for potential energy is:
Potential Energy = mass × gravitational acceleration × height
Since the mass of Tarzan is not given, we can assume it to be canceled out in the equation. In this case, we only need to calculate the change in potential energy, which is equal to the potential energy at the bottom minus the potential energy at the top. Since Tarzan starts from rest, he has no kinetic energy at the top of the swing. Therefore, we only need to calculate the potential energy at the bottom of the swing.
2. The potential energy at the bottom of the swing can be calculated using the same formula:
Potential Energy = mass × gravitational acceleration × height
However, we need to find the height at the bottom of the swing. To find the height, we can use trigonometry. The given length of the vine is the hypotenuse of a right triangle, and the angle gives us the opposite side. Using the formula:
Height = Vine length × sin(angle)
3. Once we have the potential energy at the bottom of the swing, we can convert it into kinetic energy. Since there is no additional energy input, the total mechanical energy remains the same throughout the swing.
Kinetic Energy = Potential Energy at the bottom
4. Finally, we can calculate the speed at the bottom of the swing using the formula for kinetic energy:
Kinetic Energy = 1/2 × mass × velocity^2
Rearranging the equation gives us:
velocity = √(2 × (Potential Energy at the bottom) / mass)
Please note that to find an accurate answer, we need to know the mass of Tarzan. If it is not provided, we can't calculate the exact value.
Use trignometry to compute how much higher he is at the 35 degree angular position that at the bottom of the swing. Call that distance h. You will need to subtract a cosine term from the length to get h. This is something you should be able to do.
Then use conservation of energy to get the speed at the bottom:
m g h = (1/2) m v^2
v = sqrt (2 g h)