A 22.9-kg child is on a swing attached to ropes that are L = 1.33 m long. Take the zero of the gravitational potential energy to be at the position of the child when the ropes are horizontal.

Part A)The child's gravitational potential energy when the child is at the lowest point of the circular trajectory=-298.8 J
Part B)Determine the child's gravitational potential energy when the ropes make an angle of 45.0° relative to the vertical.
and
Based on the results in part a) and b), which position has the higher potential energy?
the position in part b) has greater potential energy
the position in part a) has greater potential energy

U = m g h

A)
U = 0 - 22.9 * 9.81 * 1.33
= -298.8 J
B)
U = 0 - 22.9 * 9.81 * cos 45
= -158.9
C)
B is higher than A and has higher (less negative) potential energy.

Oh, the potential energy from physics! I'm glad you specified because I was thinking of potential energy as in the potential for the child to become a world-class comedian. Anyway, let's get to the questions.

Part A) The child's gravitational potential energy when the child is at the lowest point of the circular trajectory is -298.8 J. Well, that's not a very uplifting situation for the poor child. I hope they're not too bummed out about it!

Part B) Now let's determine the child's gravitational potential energy when the ropes make an angle of 45.0° relative to the vertical. Hmm, well, that's a bit of a twisted situation for the child, isn't it? I guess it's like they're hanging out at a diagonal angle, which must be a bit disorienting. But hey, at least they're not at their lowest point anymore!

Based on the results in part a) and b), which position has the higher potential energy? Now this is an interesting question. Drum roll, please... The position in part a) actually has greater potential energy! Can you believe it? I guess being at the lowest point of the circular trajectory creates a real energy low for the child. Time to bring in some positive vibes, people!

So there you have it, the potential energy situation for our swinging child. Remember, in the world of physics, potential energy can be a real rollercoaster ride!

To find the gravitational potential energy at different positions, we need to use the equation:

Gravitational Potential Energy (E) = mgh,

where:
m = mass of the child = 22.9 kg,
g = acceleration due to gravity = 9.8 m/s^2,
h = height (vertical distance) from the reference point.

Part A:
The child is at the lowest point of the circular trajectory, which means the height (h) is zero. Plugging in the given values:

Ea = mgh
= 22.9 kg * 9.8 m/s^2 * 0 m
= 0 J

Therefore, the child's gravitational potential energy at the lowest point is 0 J.

Part B:
The ropes make an angle of 45.0° relative to the vertical. We can find the height (h') by considering the right-angled triangle formed by the ropes and the vertical line:

sin(45°) = h'/L,

where:
L = length of the ropes = 1.33 m.

Rearranging the equation:

h' = L * sin(45°)
= 1.33 m * 0.7071
= 0.942 cm.

Plugging in the values to find the gravitational potential energy:

Eb = mgh'
= 22.9 kg * 9.8 m/s^2 * 0.942 m
≈ 206.72 J.

Therefore, the child's gravitational potential energy when the ropes make an angle of 45.0° relative to the vertical is approximately 206.72 J.

Based on the results in part A and part B, the position in part B (when the ropes make an angle of 45.0° relative to the vertical) has a higher potential energy compared to the position in part A (when the child is at the lowest point of the circular trajectory).

To determine the child's gravitational potential energy at different positions, we can use the formula:

PE = mgh

where PE represents the potential energy, m is the mass of the child, g is the acceleration due to gravity, and h is the height or vertical distance from the zero of the gravitational potential energy.

Part A) The child's gravitational potential energy at the lowest point of the circular trajectory can be calculated using the given information. The zero of the gravitational potential energy is when the ropes are horizontal. Therefore, the height (h) at the lowest point is 0.

PE = mgh
PE = (22.9 kg)(9.8 m/s^2)(0 m)
PE = 0 J

So, the child's gravitational potential energy at the lowest point is 0 J.

Part B) To determine the child's gravitational potential energy when the ropes make an angle of 45.0° relative to the vertical, we need to find the vertical height (h) at that position.

In a right-angled triangle, we can use the sine function to find the height (h) relative to the hypotenuse (L) and angle (θ):

sin(θ) = h/L

Rearranging the equation, we can solve for h:

h = L × sin(θ)

h = (1.33 m) × sin(45°)
h = 0.942 m

Now that we have the height (h), we can calculate the child's gravitational potential energy at this position:

PE = mgh
PE = (22.9 kg)(9.8 m/s^2)(0.942 m)
PE ≈ 204.4 J

Therefore, the child's gravitational potential energy when the ropes make an angle of 45.0° relative to the vertical is approximately 204.4 J.

Based on the results in part A and B, we can compare the potential energies. The potential energy at the lowest point is 0 J, while the potential energy at the position where the ropes make an angle of 45.0° relative to the vertical is 204.4 J. Therefore, the position in part B has greater potential energy.