A 100-N child is in a swing that is attached to ropes 1.90 m long. Find the gravitational potential energy (b) when the ropes make a 35.0° angle with the vertical
c) when the child is at the bottom of the circular arc
To find the gravitational potential energy (GPE) of the child when the ropes make a 35.0° angle with the vertical, we need to use the formula for GPE:
GPE = m * g * h
where:
m is the mass of the child
g is the acceleration due to gravity
h is the height from the reference point
Given that the child's weight is 100 N, we can find the mass using the formula:
weight = mass * g
Rearranging the formula, we have:
mass = weight / g
The value of g can be taken as 9.8 m/s^2, which is the approximate average acceleration due to gravity on Earth.
Now, let's find the mass of the child:
mass = 100 N / 9.8 m/s^2
mass ≈ 10.2 kg
Next, we need to determine the height, h. When the ropes make a 35.0° angle with the vertical, the height can be found using trigonometry. In this case, the height is the vertical component of the swing's length:
h = length of the swing * sin(angle)
h = 1.90 m * sin(35.0°)
h ≈ 1.09 m
Now we can calculate the gravitational potential energy:
GPE = mass * g * h
GPE ≈ 10.2 kg * 9.8 m/s^2 * 1.09 m
GPE ≈ 108.78 J
Therefore, the gravitational potential energy (b) when the ropes make a 35.0° angle with the vertical is approximately 108.78 J.
To find the gravitational potential energy (GPE) when the child is at the bottom of the circular arc, we can use the same formula:
GPE = m * g * h
The height, h, in this case, is the distance from the reference point to the bottom of the swing's circular arc. It is equal to the length of the swing minus the vertical component at the minimum point.
h = length of the swing - (length of the swing * cos(angle))
h = 1.90 m - (1.90 m * cos(35.0°))
h ≈ 1.23 m
Now we can calculate the gravitational potential energy:
GPE = mass * g * h
GPE ≈ 10.2 kg * 9.8 m/s^2 * 1.23 m
GPE ≈ 122.71 J
Therefore, the gravitational potential energy (c) when the child is at the bottom of the circular arc is approximately 122.71 J.