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Two children hang by their hands from the same tree branch. The branch is straight, and grows from the tree trunk at an angle of 27 degrees above the horizontal. One child, with a mass of 39 kg is hanging 1.50 m along the branch from the tree trunk. The other child, with a mass of 29 kg, is hanging 2.10 m from the tree trunk. What is the magnitude of the net torque (in Nm) exerted on the branch by the children. Assume that the axis of rotation is located where branch joins the tree trunk and is perpendicular to the plane formed by the branch and trunk.
To find the magnitude of the net torque exerted on the branch by the children, we need to calculate the torque exerted by each child and then add them together.
The torque exerted by an object is given by the equation:
Torque = Force * Distance * sin(θ)
In this case, we can assume that the force exerted by the children is equal to their weight, which can be calculated using the formula:
Weight = mass * gravity
The distance is the perpendicular distance from the axis of rotation to the line of action of the force. Since the branch is straight and grows from the tree trunk at an angle above the horizontal, we can use trigonometry to calculate the distance.
For the child with a mass of 39 kg hanging 1.50 m along the branch:
Weight = 39 kg * 9.8 m/s^2 (acceleration due to gravity)
Distance = 1.50 m * sin(27 degrees)
For the child with a mass of 29 kg hanging 2.10 m from the tree trunk:
Weight = 29 kg * 9.8 m/s^2
Distance = 2.10 m * sin(27 degrees)
Now, we can calculate the torque exerted by each child:
Torque_child1 = Weight_child1 * Distance_child1
Torque_child2 = Weight_child2 * Distance_child2
Finally, we can calculate the net torque by summing up the torques exerted by each child:
Net Torque = Torque_child1 + Torque_child2
Now you can plug in the values and perform the calculations to find the magnitude of the net torque exerted on the branch by the children.