Two children hang by their hands from the same tree branch. The branch is straight, and grows out from the tree at an angle of 26° above the horizontal. One child, with a mass of 38 kg, is 1.1 m along the branch from the tree trunk. The other child, with a mass of 34 kg, is 2.9 m along the branch 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 find the magnitude of the net torque exerted on the branch by the children, we need to determine the torque exerted by each child and then add them together.
The torque exerted by an object is given by the formula:
τ = r * F * sin(θ)
where
τ is the torque,
r is the distance from the axis of rotation to the point of force application,
F is the magnitude of the force, and
θ is the angle between the force and the line connecting the axis of rotation to the point of force application.
Let's calculate the torque exerted by each child:
1. For the first child with a mass of 38 kg, located 1.1 m along the branch:
The force exerted by the child's weight can be calculated using the equation F = m * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
So F1 = 38 kg * 9.8 m/s^2 = 372.4 N.
The angle θ1 is given as 26°.
Plugging the values into the torque formula, we get:
τ1 = (1.1 m) * (372.4 N) * sin(26°).
2. For the second child with a mass of 34 kg, located 2.9 m along the branch:
Similar to the first child, the force exerted by the second child's weight is F2 = 34 kg * 9.8 m/s^2 = 333.2 N.
The angle θ2 is also 26°.
Plugging the values into the torque formula, we get:
τ2 = (2.9 m) * (333.2 N) * sin(26°).
Now, we can find the magnitude of the net torque by summing the torques exerted by each child:
Net torque = |τ1| + |τ2|.
(Note: We use the absolute value since torque is a vector quantity and magnitude is always positive.)
You can now calculate the magnitude of the net torque by substituting the values into the formula.
To calculate the magnitude of the net torque exerted on the branch by the children, we need to find the torque exerted by each child and then add them together.
Torque is calculated using the formula:
τ = r * F * sin(θ)
Where:
- τ is the torque
- r is the distance from the axis of rotation to the point where the force is applied
- F is the force applied
- θ is the angle between the line connecting the axis of rotation and the point where the force is applied
Let's calculate the torque exerted by each child:
For the child with a mass of 38 kg:
r1 = 1.1 m (distance from the tree trunk)
F1 = 38 kg * 9.8 m/s² (force exerted due to the gravitational pull) = 372.4 N (weight of the child)
θ1 = 90° - 26° = 64° (angle with the horizontal)
τ1 = r1 * F1 * sin(θ1)
= 1.1 m * 372.4 N * sin(64°)
= 1.1 * 372.4 * 0.8988
≈ 439.3 N·m
For the child with a mass of 34 kg:
r2 = 2.9 m (distance from the tree trunk)
F2 = 34 kg * 9.8 m/s² (force exerted due to the gravitational pull) = 333.2 N (weight of the child)
θ2 = 90° - 26° = 64° (angle with the horizontal, same as the first child)
τ2 = r2 * F2 * sin(θ2)
= 2.9 m * 333.2 N * sin(64°)
= 2.9 * 333.2 * 0.8988
≈ 889.3 N·m
Now, let's find the magnitude of the net torque by adding the torques together:
Net torque = τ1 + τ2
= 439.3 N·m + 889.3 N·m
= 1328.6 N·m
Therefore, the magnitude of the net torque exerted on the branch by the children is approximately 1328.6 N·m.