A 40.0kg child is in a swing that is attached to ropes 2.00m long. Find the gravitational potential energy associated with the child relative to the child's lowest position under the following conditions:
a) When the ropes are horizontal
b) When the ropes make a 30.0 degree angle with the vertical
c) at the bottom of the circular arc
To find the gravitational potential energy associated with the child in each of the given conditions, we need to calculate the height at each position.
a) When the ropes are horizontal:
In this case, the height of the child is equal to the length of the ropes since the child is at the maximum height. The gravitational potential energy is given by the formula:
Gravitational Potential Energy (U) = m * g * h,
where
m = mass of the child = 40.0 kg,
g = acceleration due to gravity = 9.8 m/s²,
h = height above the reference point, which is the lowest position.
So, U = 40.0 kg * 9.8 m/s² * 2.00 m = 784 J
b) When the ropes make a 30.0 degree angle with the vertical:
In this case, the height of the child is the vertical component of the rope length. We can calculate it using the equation h = l * sin(θ), where
l = length of the ropes = 2.00 m, and
θ = angle with the vertical = 30.0 degrees.
So, h = 2.00 m * sin(30.0 degrees) = 1.00 m.
Now, we can substitute the values into the gravitational potential energy formula:
U = 40.0 kg * 9.8 m/s² * 1.00 m = 392 J
c) At the bottom of the circular arc:
At the bottom of the swing's circular arc, the gravitational potential energy is at its minimum value. The child's mass and height above the reference point are still the same, but the gravitational potential energy is zero because the child is at its lowest position.
Therefore, U = 0 J.
To summarize:
a) When the ropes are horizontal: U = 784 J
b) When the ropes make a 30.0 degree angle with the vertical: U = 392 J
c) At the bottom of the circular arc: U = 0 J
b
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m g h = 40 * 9.8 * h
how high above bottom of swing is h
a) h = 2 meters
b) h = 2 (1 - cos 30)
c) 0