A 390-N child is in a swing that is attached to a pair of ropes 1.9 m long. Find the gravitational potential energy (in JOULES) of the child–Earth system relative to the child's lowest position at the following times.
A) when the ropes are horizontal
B) when the ropes make a 28.0° angle with the vertical
C) when the child is at the bottom of the circular arc
To find the gravitational potential energy of the child-Earth system at different positions, we need to use the formula:
Gravitational Potential Energy (PE) = m * g * h
Where:
m = mass of the child
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height or vertical displacement from the lowest position
A) When the ropes are horizontal:
In this case, the vertical displacement (h) is zero because the swing is at its lowest point. Therefore, the gravitational potential energy is also zero.
B) When the ropes make a 28.0° angle with the vertical:
To find the vertical displacement (h), we need to calculate the difference between the length of the ropes and the horizontal distance between the attachment point and the swing position.
The horizontal distance is given by: h_horizontal = length of the ropes * sin(angle)
h_horizontal = 1.9 m * sin(28.0°)
h_horizontal ≈ 1.9 m * 0.46947 ≈ 0.891 m
The vertical displacement is given by: h = length of the ropes - h_horizontal
h = 1.9 m - 0.891 m = 1.009 m
Now we can calculate the gravitational potential energy:
PE = 390 N * 9.8 m/s^2 * 1.009 m
PE ≈ 3855.7 J
Therefore, the gravitational potential energy relative to the child's lowest position when the ropes make a 28.0° angle with the vertical is approximately 3855.7 Joules.
C) When the child is at the bottom of the circular arc:
In this case, the vertical displacement is equal to the length of the ropes, which is 1.9 m.
PE = 390 N * 9.8 m/s^2 * 1.9 m
PE ≈ 7222.4 J
Therefore, the gravitational potential energy relative to the child's lowest position when the child is at the bottom of the circular arc is approximately 7222.4 Joules.