A 350 N child is in a swing that is attached to a 7.50 m long rope. find the gravitational potential energy associated with the child relative to her lowest position when a) the ropes are horizontal, (b) the ropes make a 30 degree angle with the vertical, and (c) the child is at the bottom of the circular arc.

To calculate the gravitational potential energy associated with the child in each scenario, we need to consider the change in height from the lowest position. Gravitational potential energy (PE) is given by the equation:

PE = mgh

Where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s² on Earth), and h is the change in height.

a) When the ropes are horizontal:
In this case, the change in height is equal to the length of the rope, which is 7.50 m. To find the mass, we need to divide the force acting on the child (350 N) by the acceleration due to gravity (9.8 m/s²):

m = 350 N / 9.8 m/s² = 35.71 kg

Now we can calculate the gravitational potential energy:

PE = (35.71 kg)(9.8 m/s²)(7.50 m) = 2628.825 J

b) When the ropes make a 30-degree angle with the vertical:
In this case, we need to calculate the vertical component of the change in height. The vertical component can be found by multiplying the length of the rope by the sine of the angle (30 degrees):

h = (7.50 m)sin(30°) = 3.75 m

Now we can calculate the gravitational potential energy using the same equation:

PE = (35.71 kg)(9.8 m/s²)(3.75 m) = 1313.4375 J

c) When the child is at the bottom of the circular arc:
At the bottom of the circular arc, the height is zero, so the gravitational potential energy is also zero.

Therefore, the gravitational potential energy associated with the child relative to her lowest position is:
a) 2628.825 J
b) 1313.4375 J
c) 0 J