A 40kg child is swinging on a swing. At the bottom of the swinging motion, the child is 0.6m above the
ground and travelling at a speed of 5m/s. The chains the swing is attached to are 2.1m long.
a) How high above the ground is the child at the highest point of their swinging motion?
b) At the highest point of the swinging motion, what is the angle between the chains and their natural
resting position?
First, let's calculate the energy at the bottom of the swing using the principle of conservation of energy:
At the bottom of the swing:
Potential energy = m * g * h
Kinetic energy = 0.5 * m * v^2
where m = mass of the child (40 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height above the ground (0.6 m)
v = speed of the child (5 m/s)
Potential energy at the bottom = 40 kg * 9.8 m/s^2 * 0.6 m = 235.2 J
Kinetic energy at the bottom = 0.5 * 40 kg * (5 m/s)^2 = 500 J
Using the principle of conservation of energy, the total energy at the highest point of the swing will be equal to the total energy at the bottom:
Potential energy at the highest point + Kinetic energy at the highest point = Potential energy at the bottom + Kinetic energy at the bottom
The total energy at the highest point is the potential energy because the kinetic energy is zero when the child reaches the highest point.
Therefore:
Potential energy at the highest point = 235.2 J
To calculate the height above the ground at the highest point, we can rearrange the potential energy formula:
Potential energy = m * g * h
h = Potential energy / (m * g)
h = 235.2 J / (40 kg * 9.8 m/s^2)
h ≈ 0.6 m
Therefore, the child is approximately 0.6 m above the ground at the highest point of their swinging motion.
Now let's calculate the angle between the chains and their natural resting position at the highest point. At the highest point, the velocity is zero, so the child is momentarily at rest. The angle between the chains and their natural resting position is the same as the angle between the chains and the vertical line.
Using trigonometry, we can find this angle:
sin(angle) = opposite/hypotenuse
sin(angle) = height above the ground/length of the chains
angle = arcsin(height above the ground/length of the chains)
angle = arcsin(0.6 m/2.1 m)
angle ≈ 16.5 degrees
Therefore, at the highest point of the swinging motion, the angle between the chains and their natural resting position is approximately 16.5 degrees.