A father pulls his son, whos mass is 25 kg and who is sitting on a swing with ropes of length 3 meters, backward until the ropes make an angle of 33.6 degrees with respect to the vertical. He then releases his son from rest. What is the speed of the son at the bottom of the swinging motion?
the answer is wrong
it will equal = 3.135
Well, let's swing into action and do some math! To find the speed of the son at the bottom of the swinging motion, we can use the conservation of mechanical energy. At the top of the swing, all of the potential energy is converted into kinetic energy.
First, let's find the potential energy at the top of the swing. The potential energy is given by the formula:
Potential Energy = mass * g * height
We can calculate the height of the swing by considering it as an isosceles triangle. The angle at the top is 33.6 degrees, so the perpendicular height (h) can be found using basic trigonometry - h = length of ropes * sin(angle).
Now that we have the potential energy at the top, we can calculate the kinetic energy at the bottom of the swing using the conservation of energy. The kinetic energy is given by the formula:
Kinetic Energy = 0.5 * mass * velocity^2
Since the potential energy is converted into kinetic energy, we can equate the two:
Potential Energy at the top = Kinetic Energy at the bottom
mass * g * height = 0.5 * mass * velocity^2
Simplifying the equation, we find:
velocity = sqrt(2 * g * height)
Substituting the values, we get:
velocity = sqrt(2 * 9.8 * (3 * sin(33.6)))
Now it's time for the punchline! Let's calculate the speed:
velocity = sqrt(2 * 9.8 * (3 * 0.5597))
velocity ≈ 8.68 m/s
So, the speed of the son at the bottom of the swinging motion is approximately 8.68 meters per second.
To find the speed of the son at the bottom of the swinging motion, we can use the principle of conservation of mechanical energy. At the top of the swing, all the potential energy is converted into kinetic energy.
1. First, let's find the potential energy at the top of the swing:
- Mass of the son (m) = 25 kg
- Gravitational acceleration (g) = 9.8 m/s^2
- Height (h) = length of the ropes = 3 meters
Potential energy (PE) = m * g * h
PE = 25 kg * 9.8 m/s^2 * 3 m
PE = 735 Joules (J)
2. At the bottom of the swing, all the potential energy is converted into kinetic energy.
- Kinetic energy (KE) = Potential energy (PE) = 735 J
KE = (1/2) * m * v^2
735 J = (1/2) * 25 kg * v^2
3. Solve for the velocity (v):
v^2 = (2 * 735 J) / 25 kg
v^2 = 58.8 m^2/s^2
Taking the square root of both sides:
v = √(58.8 m^2/s^2)
v = 7.66 m/s
Therefore, the speed of the son at the bottom of the swinging motion is 7.66 m/s.
To find the speed of the son at the bottom of the swinging motion, we can use the principle of conservation of mechanical energy.
Here's how you can calculate it step by step:
1. Identify the gravitational potential energy and the kinetic energy at the highest point and bottom of the swing.
At the highest point:
- The potential energy is at its maximum.
- The kinetic energy is zero because the son is momentarily at rest.
At the bottom of the swing:
- The potential energy is zero because it's the lowest point.
- The kinetic energy is at its maximum.
2. Calculate the potential energy at the highest point:
The potential energy (PE) is given by the formula: PE = m * g * h
Where:
- m is the mass of the son (25 kg).
- g is the acceleration due to gravity (approximately 9.8 m/s²).
- h is the height (vertical distance) from the highest point to the rest position.
Since the ropes are 3 meters long, and the angle made with the vertical is 33.6 degrees, the height can be calculated using trigonometry:
h = 3 * sin(33.6°)
3. Calculate the potential energy at the highest point:
PE = m * g * h
4. Calculate the kinetic energy at the bottom of the swing:
At the bottom of the swing, all the potential energy has been converted into kinetic energy, which is given by the formula: KE = 1/2 * m * v²
Where:
- m is the mass of the son (25 kg).
- v is the velocity (speed) of the son at the bottom of the swing.
Since both potential energy and kinetic energy are equivalent, we can set them equal to each other:
PE = KE
m * g * h = 1/2 * m * v²
5. Solve for the velocity (v):
Divide both sides of the equation by (1/2 * m):
2 * m * g * h = m * v²
Simplify:
2 * g * h = v²
Take the square root of both sides:
v = √(2 * g * h)
6. Substitute the values and calculate the velocity (v):
Substitute the values for g (9.8 m/s²) and h (3 * sin(33.6°)) into the equation and calculate v.
v = √(2 * 9.8 * 3 * sin(33.6°))
Using a calculator, evaluate the expression to find the velocity.
Note: Make sure to convert the angle from degrees to radians if your calculator requires it.
That's it! Follow these steps, and you will find the velocity of the son at the bottom of the swinging motion.
How high?
Draw it.
h = L - L cos 33.6 = L(1-cos33.6)
= 3(1-.833) = .501 meters up
Potential energy increase = m g h
= 25 (9.81) (.501)
Ke = (1/2)(25) v^2 = 25(9.81)(.501)
cancel the mass and
solve for v = sqrt(2 g h) as usual :)