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?
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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 mechanical energy at the top of the swing (when the son is pulled back) is equal to the mechanical energy at the bottom of the swing (when the son is at its lowest point).
The mechanical energy is the sum of the kinetic energy (KE) and the potential energy (PE). At the top of the swing, all of the son's energy is in potential energy, and at the bottom of the swing, all of the energy is in kinetic energy.
First, we need to calculate the potential energy at the top of the swing. The potential energy (PE) is given by the formula:
PE = m * g * h
Where:
m = mass of the son = 25 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height of the swing from the lowest point to the top = 3 m * (1 - cos(33.6))
Next, we calculate the kinetic energy at the bottom of the swing. The kinetic energy (KE) is given by the formula:
KE = 1/2 * m * v^2
Where:
m = mass of the son = 25 kg
v = velocity at the bottom of the swing (what we want to find)
According to the conservation of mechanical energy, the potential energy at the top of the swing is equal to the kinetic energy at the bottom of the swing:
PE(top) = KE(bottom)
m * g * h = 1/2 * m * v^2
Now, we can solve for v to find the speed of the son at the bottom of the swing:
v = sqrt((2 * g * h))
Substituting the values of g and h into the equation, we have:
v = sqrt((2 * 9.8 * 3 * (1 - cos(33.6))))
Calculating this expression will give us the speed of the son at the bottom of the swinging motion.