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 27.3° above the horizontal. One child, with a mass of 40.8 kg, is hanging 1.24 m along the branch from the tree trunk. The other child, with a mass of 36.3 kg, is hanging 2.09 m from the tree trunk. What is the magnitude of the net torque exerted on the branch by the children?

Question

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 27.3° above the horizontal. One child, with a mass of 40.8 kg, is hanging 1.24 m along the branch from the tree trunk. The other child, with a mass of 36.3 kg, is hanging 2.09 m from the tree trunk. What is the magnitude of the net torque exerted on the branch by the children?

I would 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.

Answer 1

To find 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 add them together.

Torque is given by the formula:
τ = r × F

where τ is the torque, r is the lever arm (perpendicular distance between the axis of rotation and the line of action of force), and F is the force.

Let's calculate the torque exerted by each child:

For the first child (with a mass of 40.8 kg) hanging 1.24 m along the branch from the tree trunk:

The force F1 can be calculated using the formula F1 = m1 × g, where m1 is the mass of the first child and g is the acceleration due to gravity (9.8 m/s²).

F1 = 40.8 kg × 9.8 m/s² = 400.56 N

The lever arm r1 is the perpendicular distance between the axis of rotation and the line of action of force. Since the line of action of force is along the branch, r1 is the horizontal distance.

r1 = 1.24 m × cos(27.3°)
= 1.24 m × 0.8916
= 1.104 m

Therefore, the torque exerted by the first child (τ1) is:
τ1 = r1 × F1

For the second child (with a mass of 36.3 kg) hanging 2.09 m from the tree trunk:

The force F2 can be calculated using the same formula: F2 = m2 × g

F2 = 36.3 kg × 9.8 m/s² = 355.74 N

The lever arm r2 is also the horizontal distance:
r2 = 2.09 m × cos(27.3°)
= 2.09 m × 0.8916
= 1.863 m

Therefore, the torque exerted by the second child (τ2) is:
τ2 = r2 × F2

Finally, to find the net torque (τ_net), we add the torques exerted by both children:
τ_net = τ1 + τ2

So, to get the magnitude of the net torque exerted on the branch by the children, you can calculate τ_net using the formulas mentioned above.