A father pulls his son, whose mass is 27.3 kg and who is sitting on a swing with ropes of length 2.85 m, backward until the ropes make an angle of 36.8° 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?
To find the speed of the son at the bottom of the swinging motion, we can use conservation of mechanical energy.
Step 1: Find the gravitational potential energy at the top of the swing.
Using the formula for gravitational potential energy, we can calculate the potential energy at the top of the swing:
Potential Energy = mgh
where m is the mass, g is the acceleration due to gravity, and h is the height.
Given:
Mass of the son (m) = 27.3 kg
Acceleration due to gravity (g) = 9.8 m/s^2 (approximate value)
Height (h) = length of the ropes * (1 - cos(angle))
Using trigonometry, we can find the height:
Height = 2.85 m * (1 - cos(36.8°))
Calculate the height.
Step 2: Convert potential energy to kinetic energy at the bottom of the swing.
At the bottom of the swing, all potential energy is converted to kinetic energy.
Therefore, Kinetic Energy = Potential Energy at the top of the swing.
Solve for the kinetic energy.
Step 3: Find the speed at the bottom of the swing.
Using the formula for kinetic energy:
Kinetic Energy = 1/2 * mv^2
where m is the mass and v is the velocity.
Solve for the velocity by rearranging the formula.
Step 4: Calculate the speed.
Plug in the values obtained into the formula and calculate the speed.
Let's calculate the speed of the son at the bottom of the swing.
To find the speed of the son at the bottom of the swinging motion, we need to consider the conservation of mechanical energy.
The mechanical energy of the system is conserved, which means that the sum of kinetic energy and potential energy remains constant throughout the swing. At the start, when the son is at the highest point (pulling backward), all of the energy is in the form of potential energy. At the bottom of the swing, all of the energy is in the form of kinetic energy.
The potential energy at the highest point is given by the formula:
PE = mgh
where m is the mass of the son, g is the acceleration due to gravity, and h is the height of the swing.
In this case, the height of the swing is the vertical distance between the starting position and the lowest point of the swing, which can be calculated using the length of the swing ropes and the angle made by the ropes with the vertical:
h = L(1 - cosθ)
where L is the length of the ropes and θ is the angle made by the ropes with the vertical.
The kinetic energy at the bottom of the swing can be calculated using the formula:
KE = 0.5mv^2
where v is the velocity of the son at the bottom of the swing.
Since the mechanical energy is conserved, we can equate the initial potential energy to the final kinetic energy:
PE = KE
mgh = 0.5mv^2
Now, let's plug in the given values:
m = 27.3 kg
L = 2.85 m
θ = 36.8° (converted to radians by multiplying by π/180)
g = 9.8 m/s^2
First, calculate the height of the swing:
h = L(1 - cosθ)
h = 2.85(1 - cos(36.8°))
h = 1.86 m
Now, equate the initial potential energy to the final kinetic energy:
mgh = 0.5mv^2
27.3 kg * 9.8 m/s^2 * 1.86 m = 0.5 * 27.3 kg * v^2
Simplifying the equation:
267.0864 kg·m^2/s^2 = 13.65 kg * v^2
Dividing both sides by 13.65 kg:
v^2 = 19.5379 m^2/s^2
Taking the square root of both sides:
v = 4.42 m/s
So, the speed of the son at the bottom of the swinging motion is approximately 4.42 m/s.