Two children hang by their hands from the same tree branch. The branch is straight, and grows out from the tree trunk at an angle of 26.1° above the horizontal. One child, with a mass of 43.6 kg, is hanging 1.31 m along the branch from the tree trunk. The other child, with a mass of 32.7 kg, is hanging 1.89 m from the tree trunk. What is the magnitude of the net torque exerted on the branch by the children? Assume that the axis is located where the branch joins the tree trunk and is perpendicular to the plane formed by the branch and the trunk.

To calculate the magnitude of the net torque exerted on the branch by the children, we need to first calculate the torque exerted by each child and then find the sum of the torques.

The torque exerted by an object is given by the formula:
τ = F * r * sin(θ)

Where:
τ is the torque
F is the force applied
r is the distance from the axis of rotation
θ is the angle between the force vector and the line from the axis to the point of application of the force

For the first child, the force F is equal to the weight of the child, which can be calculated as:
F1 = m1 * g
where m1 is the mass of the first child and g is the acceleration due to gravity.

Similarly, for the second child:
F2 = m2 * g

Now, we can calculate the torque exerted by each child:
τ1 = F1 * r1 * sin(θ)
τ2 = F2 * r2 * sin(θ)

Given:
m1 = 43.6 kg
m2 = 32.7 kg
r1 = 1.31 m
r2 = 1.89 m
θ = 26.1°

First, we need to convert the angle from degrees to radians:
θ_radians = θ * π / 180

Next, let's calculate the torque exerted by each child:

F1 = m1 * g
F2 = m2 * g

Now, plug in the values and calculate the torques:

τ1 = F1 * r1 * sin(θ_radians)
τ2 = F2 * r2 * sin(θ_radians)

Then, find the net torque by adding the individual torques:

net torque = τ1 + τ2

Finally, calculate the magnitude of the net torque by taking the absolute value of the net torque:

magnitude of net torque = |net torque|

Note: Don't forget to consider the direction of the torque. The torque may be positive or negative depending on the orientation of the force vector.