A 120 N child is in a swing that is attached to ropes 2.20 m long. Find the gravitational potential energy 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 29.0° angle with the vertical
(c) when the child is at the bottom of the circular arc
(a) =F*h
120*2.2= 264
(b)264*(1-cos29)
=33.1
(c)zero
(a) M*g*R
(b) M*g*R (1 - cos 29)
(c) zero
In this case, M g = 120 N
(a) 264
(b) 461.49
(c) 0
To find the gravitational potential energy of the child-Earth system at different positions, we need to consider the change in potential energy relative to the child's lowest position. The formula to calculate gravitational potential energy is:
Potential energy (PE) = mass (m) * gravity (g) * height (h)
In this case, the child's mass is not given, but we can use the weight of the child to find the mass through the formula:
Weight (W) = mass (m) * gravity (g)
Given that the weight of the child is 120 N, we can find the mass using the equation:
120 N = m * 9.8 m/s^2 (acceleration due to gravity)
Solving for m, we find:
m = 120 N / 9.8 m/s^2 ≈ 12.24 kg
Now, we can proceed to calculate the gravitational potential energy at each given position.
(a) When the ropes are horizontal:
In this case, the height (h) is the vertical distance between the child's lowest position and the current position, which is 2.20 m.
PE = m * g * h
= 12.24 kg * 9.8 m/s^2 * 2.20 m
≈ 266.06 J
Therefore, the gravitational potential energy of the child-Earth system when the ropes are horizontal is approximately 266.06 Joules.
(b) When the ropes make a 29.0° angle with the vertical:
To find the height (h), we can use the length of the ropes and the angle with the vertical. The vertical height (h) can be calculated using the equation:
h = length of ropes * sin(angle with vertical)
h = 2.20 m * sin(29.0°)
h ≈ 1.17 m
PE = m * g * h
= 12.24 kg * 9.8 m/s^2 * 1.17 m
≈ 135.01 J
Therefore, the gravitational potential energy of the child-Earth system when the ropes make a 29.0° angle with the vertical is approximately 135.01 Joules.
(c) When the child is at the bottom of the circular arc:
At the bottom of the circular arc, the height (h) is 0 since it is the child's lowest position.
PE = m * g * h
= 12.24 kg * 9.8 m/s^2 * 0 m
= 0 J
Therefore, the gravitational potential energy of the child-Earth system at the bottom of the circular arc is 0 Joules.