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 33° above the horizontal. One child, with a mass of 51 kg, is 1.4 m along the branch from the tree trunk. The other child, with a mass of 26 kg, is 2.1 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.
τ =m₁gL₁cosα +m₂gL₂cosα =
= gcosα(m₁L₁ +m₂L₂) =
=9.8•cos33°(51•1.4+26•2.1) =
=9.8•0.84•(71.4+54.6)=1037.232 N•m
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 is the torque, r is the 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 lever arm.
For the first child with a mass of 51 kg, the force exerted by the child is equal to the weight of the child, which can be calculated using the equation:
F = m * g
where m is the mass of the child and g is the acceleration due to gravity (approximately 9.8 m/s²).
Substituting the values for the first child:
m = 51 kg
g = 9.8 m/s²
F = 51 kg * 9.8 m/s² = 499.8 N
The distance from the axis of rotation to the first child is given as 1.4 m.
Substituting the values into the torque equation:
τ₁ = 1.4 m * 499.8 N * sin(33°)
Next, we calculate the torque exerted by the second child:
m = 26 kg
g = 9.8 m/s²
F = 26 kg * 9.8 m/s² = 254.8 N
The distance from the axis of rotation to the second child is given as 2.1 m.
τ₂ = 2.1 m * 254.8 N * sin(33°)
Finally, we can calculate the net torque by adding the torques exerted by both children:
net torque = τ₁ + τ₂
After calculating the values, you can sum up the torques to find 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 can use the equation:
τ = r * F * sinθ
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 force and the lever arm
First, we need to calculate the torque exerted by each child, and then add them together to find the net torque.
For the first child with a mass of 51 kg and a distance of 1.4 m from the tree trunk:
- The force exerted by the child can be calculated using the formula F = m * g, where g is the acceleration due to gravity (approximately 9.8 m/s²).
F1 = (51 kg) * (9.8 m/s²) = 499.8 N
- The angle θ is given as 33°. However, we need to convert it to radians before proceeding with the calculation. θ_radian = θ_degree * π/180
θ_radian1 = 33° * (π/180) = 0.575958653 radians
- The torque exerted by the first child can be calculated using the formula:
τ1 = r1 * F1 * sin(θ_radian1)
τ1 = (1.4 m) * (499.8 N) * sin(0.575958653 radians) = 427.923 N·m
For the second child with a mass of 26 kg and a distance of 2.1 m from the tree trunk:
- The force exerted by the child is:
F2 = (26 kg) * (9.8 m/s²) = 254.8 N
- The angle θ is still 33°, and we convert it to radians:
θ_radian2 = 33° * (π/180) = 0.575958653 radians
- The torque exerted by the second child is calculated as:
τ2 = r2 * F2 * sin(θ_radian2)
τ2 = (2.1 m) * (254.8 N) * sin(0.575958653 radians) = 333.674 N·m
Finally, to find the net torque exerted by the children, we need to add the individual torques together:
Net Torque = τ1 + τ2 = 427.923 N·m + 333.674 N·m = 761.597 N·m
Therefore, the magnitude of the net torque exerted on the branch by the children is 761.597 N·m.