A father pulls his son, whose mass is 26.7 kg and who is sitting on a swing with ropes of length 3.27 m, backward until the ropes make an angle of 35.6° 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?
m g h = (1/2) m v^2
v^2 = 2 g h
v = sqrt (2 g h)
h = 3.27 (1 - cos 35.6) meters
g = 9.8 m/s^2
Well, swinging is a bit like riding a roller coaster. You start up high, experience some thrills and spills, and eventually end up at the bottom. Now, let me take a swing at answering your question!
To find the speed of the son at the bottom of the swinging motion, we can make use of the law of conservation of energy. At the highest point (when the ropes are at an angle of 35.6° with respect to the vertical), all of the son's potential energy will be converted into kinetic energy.
The potential energy of the son at the highest point is given by:
Potential Energy = mass * gravity * height
Now, the height can be calculated using the length of the ropes and the angle they make. We can use a little bit of trigonometry here:
Height = length * sin(angle)
Plug in the values for length (3.27 m) and angle (35.6°), and you can calculate the potential energy.
Then, at the bottom of the swing, the potential energy will be completely converted into kinetic energy:
Kinetic Energy = (1/2) * mass * velocity^2
Since the potential energy at the highest point is equal to the kinetic energy at the bottom, you can set up an equation:
Potential Energy = Kinetic Energy
mass * gravity * height = (1/2) * mass * velocity^2
Now, you can solve this equation to find the velocity at the bottom of the swing. Just plug in the values for mass, gravity, and the height calculated earlier, and you'll get your answer!
Remember, though, this calculation assumes no friction or air resistance, so it may not reflect the exact speed in real-life swing situations. But hey, at least you got a swingin' explanation, right?
To find the speed of the son at the bottom of the swinging motion, we can use the principle of conservation of mechanical energy.
The total mechanical energy of the system (father + son + swing) is conserved throughout the motion. At the highest point of the swing, all the gravitational potential energy is converted into kinetic energy. Therefore, we can equate the potential energy at the highest point with the kinetic energy at the bottom.
The potential energy at the highest point is given by:
PE = mgh
Where:
m = mass of the son = 26.7 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height above the bottom of the swing = 2 * length of the ropes * (1 - cosθ)
θ = angle made by the ropes with respect to the vertical = 35.6°
The potential energy can be calculated as follows:
h = 2 * 3.27 * (1 - cos(35.6°))
h = 2 * 3.27 * (1 - cos(0.6224))
h ≈ 2 * 3.27 * (1 - 0.7830)
h ≈ 2 * 3.27 * 0.2170
h ≈ 1.422 m
Substituting the values back into the potential energy equation:
PE = mgh
PE = (26.7 kg) * (9.8 m/s^2) * (1.422 m)
PE ≈ 370.422 J
Since the mechanical energy is conserved, the kinetic energy at the bottom of the swing will be equal to the potential energy at the highest point:
KE = PE
0.5mv^2 = 370.422 J
Now we can solve for the speed (v):
v^2 = (2 * PE) / m
v^2 = (2 * 370.422 J) / 26.7 kg
v^2 ≈ 27.72 m^2/s^2
Taking the square root of both sides:
v ≈ √(27.72 m^2/s^2)
v ≈ 5.27 m/s
Therefore, the speed of the son at the bottom of the swinging motion is approximately 5.27 m/s.
To find the speed of the son at the bottom of the swinging motion, we can use the concept of conservation of mechanical energy.
The total mechanical energy of the system is conserved, which means that the initial mechanical energy is equal to the final mechanical energy.
At the highest point of the swinging motion, the son's potential energy is at its maximum, while his kinetic energy is zero. At the bottom of the swing, his potential energy is zero, and his kinetic energy is at its maximum.
First, let's calculate the initial potential energy of the son at the highest point of the swing.
Potential energy (PE) = mass (m) × gravitational acceleration (g) × height (h)
Since the son is at the highest point of the swing, the height is the length of the ropes, h = 3.27 m.
Mass (m) = 26.7 kg
Gravitational acceleration (g) = 9.8 m/s^2
PE_initial = 26.7 kg × 9.8 m/s^2 × 3.27 m
Next, let's calculate the final kinetic energy of the son at the bottom of the swing.
Kinetic energy (KE) = 0.5 × mass (m) × velocity^2 (v^2)
At the bottom of the swing, the son's potential energy is zero, so all of the initial potential energy is converted into kinetic energy.
Therefore, PE_initial = KE_final
Now, substitute the values into the equation:
26.7 kg × 9.8 m/s^2 × 3.27 m = 0.5 × 26.7 kg × v^2
Simplify the equation:
(26.7 kg × 9.8 m/s^2 × 3.27 m) / (0.5 × 26.7 kg) = v^2
Calculate the right-hand side of the equation:
10.91 m^2/s^2 = v^2
To find v, the speed of the son at the bottom of the swing, take the square root of both sides:
v = √10.91 m^2/s^2
v ≈ 3.3 m/s
Therefore, the speed of the son at the bottom of the swinging motion is approximately 3.3 m/s.