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 23° above the horizontal. One child, with a mass of 59 kg, is 1.8 m along the branch from the tree trunk. The other child, with a mass of 49 kg, is 2.3 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 and then add them together.
The torque exerted by an object is given by the equation:
τ = r × F × sin(θ)
Where:
τ = torque
r = distance from the axis of rotation to the point where the force is applied
F = applied force
θ = angle between the direction of the applied force and the line connecting the axis of rotation to the point where the force is applied
Let's calculate the torque exerted by the first child:
For the first child:
Mass (m1) = 59 kg
Distance from the axis (r1) = 1.8 m
Angle with the horizontal (θ1) = 23°
The force exerted by the first child is equal to the weight (mg), where g is the acceleration due to gravity (9.8 m/s²). Therefore:
F1 = m1 * g = 59 kg * 9.8 m/s²
Now, we can calculate the torque (τ1) exerted by the first child:
τ1 = r1 * F1 * sin(θ1)
Next, let's calculate the torque exerted by the second child:
For the second child:
Mass (m2) = 49 kg
Distance from the axis (r2) = 2.3 m
Angle with the horizontal (θ2) = 23°
The force exerted by the second child is equal to the weight (mg), where g is the acceleration due to gravity (9.8 m/s²). Therefore:
F2 = m2 * g = 49 kg * 9.8 m/s²
Now, we can calculate the torque (τ2) exerted by the second child:
τ2 = r2 * F2 * sin(θ2)
Finally, we can calculate the magnitude of the net torque exerted on the branch by adding the torques exerted by each child:
Magnitude of net torque = |τ1 + τ2|
Substituting the values into the equations above and evaluating them will provide the final answer.
To find the magnitude of the net torque exerted on the branch by the children, we need to calculate the torque produced by each child and then sum them up.
Torque (τ) is given by the formula: τ = r * F * sin(θ), where r is the distance from the pivot point (axis) to the point where the force is applied, F is the magnitude of the force, and θ is the angle between the force and the line connecting the pivot to the point where the force is applied.
Let's calculate the torque produced by the first child:
The distance of the first child from the axis is 1.8 m along the branch. The force acting on the first child is the weight of the child, which can be calculated using the equation: F = m * g, where m is the mass and g is the acceleration due to gravity.
F1 = (mass of first child) * g
= 59 kg * 9.8 m/s²
= 578.2 N
The angle between the force and the branch (θ1) is 23°. Therefore, the torque produced by the first child is:
τ1 = r1 * F1 * sin(θ1)
= 1.8 m * 578.2 N * sin(23°)
≈ 231.1 N·m (rounded to 3 significant figures)
Similarly, we can calculate the torque produced by the second child:
The distance of the second child from the axis is 2.3 m along the branch. Using the same formula for force, we have:
F2 = (mass of second child) * g
= 49 kg * 9.8 m/s²
= 480.2 N
The angle between the force and the branch (θ2) is also 23°.
τ2 = r2 * F2 * sin(θ2)
= 2.3 m * 480.2 N * sin(23°)
≈ 197.0 N·m (rounded to 3 significant figures)
Finally, the net torque is the sum of the torques produced by each child:
Net torque = τ1 + τ2
= 231.1 N·m + 197.0 N·m
≈ 428.1 N·m (rounded to 3 significant figures)
Therefore, the magnitude of the net torque exerted on the branch by the children is approximately 428.1 N·m.