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 calculate the torque exerted by each child individually and then add them together.
The torque exerted by an object can be calculated using the formula:
τ = r * F * sin(θ)
Where:
τ is the torque,
r is the perpendicular distance from the axis of rotation to the point where the force is applied,
F is the magnitude of the force, and
θ is the angle between the force vector and the line connecting the axis of rotation to the point where the force is applied.
Let's start by calculating the torque exerted by the first child (child A with a mass of 38 kg) using the given information:
- The angle between the branch and the horizontal is 26°.
- The distance of child A from the tree trunk is 1.1 m.
Calculating the torque exerted by child A:
τA = rA * FA * sin(θA)
θA = 26° (angle between the branch and the horizontal)
rA = 1.1 m (distance of child A from the tree trunk)
FA = m * g (mass of child A multiplied by acceleration due to gravity)
FA = 38 kg * 9.8 m/s² (taking g as approximately 9.8 m/s²)
Now we can calculate the torque exerted by child A:
τA = 1.1 m * (38 kg * 9.8 m/s²) * sin(26°)
Next, let's calculate the torque exerted by the second child (child B with a mass of 34 kg) using the given information:
- The angle between the branch and the horizontal is 26°.
- The distance of child B from the tree trunk is 2.9 m.
Calculating the torque exerted by child B:
τB = rB * FB * sin(θB)
θB = 26° (angle between the branch and the horizontal)
rB = 2.9 m (distance of child B from the tree trunk)
FB = m * g (mass of child B multiplied by acceleration due to gravity)
FB = 34 kg * 9.8 m/s² (taking g as approximately 9.8 m/s²)
Now we can calculate the torque exerted by child B:
τB = 2.9 m * (34 kg * 9.8 m/s²) * sin(26°)
Finally, to find the magnitude of the net torque, we add the torques exerted by each child:
Magnitude of the net torque = |τA| + |τB|
Note: The absolute value (| |) ensures that we take into account the direction of the torque but consider only the magnitude.
Now you can calculate the values and find the magnitude of the net torque exerted on the branch by the children.
To find the magnitude of the net torque exerted on the branch by the children, we need to calculate the torque caused by each child and then add them together.
Torque is given by the formula: Torque = Force * Distance * sin(angle)
First, let's find the torque caused by the child with a mass of 38 kg:
1. The distance from the child to the axis is 1.1 m.
2. The angle between the branch and the horizontal is 26°.
3. The force exerted by the child can be calculated using the formula: Force = mass * acceleration due to gravity.
Assuming the acceleration due to gravity is 9.8 m/s^2:
Force = 38 kg * 9.8 m/s^2 = 372.4 N.
4. Now, we can find the torque caused by this child:
Torque = Force * Distance * sin(angle) = 372.4 N * 1.1 m * sin(26°) = 138.71 N·m (rounded to two decimal places).
Next, let's find the torque caused by the child with a mass of 34 kg:
1. The distance from the child to the axis is 2.9 m.
2. The angle between the branch and the horizontal is 26°.
3. The force exerted by the child can be calculated using the formula: Force = mass * acceleration due to gravity.
Assuming the acceleration due to gravity is 9.8 m/s^2:
Force = 34 kg * 9.8 m/s^2 = 333.2 N.
4. Now, we can find the torque caused by this child:
Torque = Force * Distance * sin(angle) = 333.2 N * 2.9 m * sin(26°) = 264.41 N·m (rounded to two decimal places).
Finally, we can find the net torque by adding the two torques together:
Net Torque = Torque_child1 + Torque_child2 = 138.71 N·m + 264.41 N·m = 403.12 N·m (rounded to two decimal places).
Therefore, the magnitude of the net torque exerted on the branch by the children is 403.12 N·m.
Torque= weight*length*sinAngle
where Angle here is the angle between the tree branch and the vertical weight. (which is 90-26)
figure the two torques, and add.