370-N child is in a swing that is attached to a pair of ropes 1.80 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 32.0° angle with the vertical
(c) when the child is at the bottom of the circular arc
To find the gravitational potential energy, we can use the formula:
Gravitational potential energy (PE) = mgh
Where:
m = mass of the child
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height above the child's lowest position
(a) When the ropes are horizontal:
In this case, the height (h) is equal to the length of the ropes (1.80 m).
PE = mgh
PE = m * 9.8 * 1.80
PE = 17.64 m * g
(b) When the ropes make a 32.0° angle with the vertical:
To find the height, we can use the vertical component of the rope's length:
h = length of the ropes * sin(angle)
h = 1.80 * sin(32.0°)
PE = mgh
PE = m * 9.8 * h
PE = 9.8 * m * 1.80 * sin(32.0°)
(c) When the child is at the bottom of the circular arc:
In this case, the height (h) is zero since the child is at the lowest position. Therefore, the gravitational potential energy is also zero.
PE = 0
Note: To find the gravitational potential energy in terms of the child's mass (m), we need to know the mass of the child.
To find the gravitational potential energy of the child-Earth system, we can use the formula:
Gravitational Potential Energy (PE) = m * g * h
Where:
- m is the mass of the child
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- h is the height of the child from the child's lowest position.
(a) When the ropes are horizontal:
In this case, the height of the child from the child's lowest position is zero. Therefore, the gravitational potential energy would also be zero.
(b) When the ropes make a 32.0° angle with the vertical:
To calculate the height (h) of the child from the child's lowest position, we need to find the vertical component (y) of the rope:
y = 1.80 m * sin(32.0°)
Next, we can calculate the height (h) of the child:
h = 1.80 m - y
Finally, we can calculate the gravitational potential energy (PE):
PE = m * g * h
(c) When the child is at the bottom of the circular arc:
In this case, the height of the child from the child's lowest position is the length of the ropes (1.80 m). We can calculate the gravitational potential energy using the same formula:
PE = m * g * h
It's important to note that we need to know the mass of the child to calculate the gravitational potential energy accurately.
To find the gravitational potential energy of the child-Earth system, we can use the formula:
Gravitational Potential Energy (PE) = mgh
where m is the mass of the child, g is the acceleration due to gravity, and h is the height relative to the child's lowest position.
(a) When the ropes are horizontal:
In this case, the child is at the highest position of the swing's arc. Therefore, the height (h) is equal to 1.80 m (the length of the ropes). The angle between the ropes and the vertical (θ) is 90°.
Using the formula, we have:
PE(a) = mgh(a) = m * g * h
(b) When the ropes make a 32.0° angle with the vertical:
In this case, we need to calculate the new height (h) of the child relative to the lowest position. To do this, we can use the trigonometric functions sine and cosine.
cos(θ) = adjacent / hypotenuse
cos(32.0°) = x / 1.80 m
x = 1.80 m * cos(32.0°)
sin(θ) = opposite / hypotenuse
sin(32.0°) = h / 1.80 m
h = 1.80 m * sin(32.0°)
Using the formula again, we have:
PE(b) = mgh(b) = m * g * h
(c) When the child is at the bottom of the circular arc:
In this case, the child's height (h) is equal to zero, as it is at the lowest position. Therefore, the gravitational potential energy is also zero.
Using these equations, you can calculate the gravitational potential energy (PE) for each scenario by plugging in the necessary values for mass (m), acceleration due to gravity (g), and the corresponding height (h).