A large arm model was constructed with a free arm which weights 1 kg and has a center of gravity 12 cm from the pivot.
a. if the force cord is attached 1 cm from the pivot, what force must be applied to support the free arm at 90 degrees?
b. an additional weight of 1 kg is supported by the arm at a distance of 12 cm from the pivot. What is the total torque of the resistance? What effort force will be required to support the arm and weight?
To solve these problems, we can use the equation for torque: torque = force × distance. Torque is the measure of the turning force around an axis, and it depends on the force applied and the distance from the axis of rotation.
a. To find the force required to support the free arm at 90 degrees (perpendicular to the ground), we need to calculate the torque required to balance the weight of the free arm.
The weight of the free arm is 1 kg, and its center of gravity is 12 cm from the pivot. Since the force cord is attached 1 cm from the pivot and we need to balance the arm at 90 degrees, the distance from the pivot to the force cord is 11 cm (12 cm - 1 cm).
We can calculate the torque required using the equation torque = force × distance. In this case, the torque required to balance the free arm is:
torque = (weight of the arm) × (distance from the force cord to the pivot)
= 1 kg × 0.11 m (converting 11 cm to meters)
= 0.11 Nm
Since torque = force × distance, we can rearrange the equation to solve for force:
force = torque / distance from the force cord to the pivot
Plugging in the values, we get:
force = 0.11 Nm / 0.01 m (converting 1 cm to meters)
= 11 N
Therefore, a force of 11 Newtons must be applied to support the free arm at 90 degrees.
b. In this case, an additional weight of 1 kg is added at a distance of 12 cm from the pivot. We need to calculate the total torque of the resistance and the effort force required to support the arm and the weight.
The total torque of the resistance is the sum of the torque caused by the weight of the free arm and the torque caused by the additional weight.
The torque caused by the weight of the free arm is given by:
torque1 = (weight of the arm) × (distance from the center of gravity of the arm to the pivot)
= 1 kg × 0.12 m (converting 12 cm to meters)
= 0.12 Nm
The torque caused by the additional weight is given by:
torque2 = (weight of the additional weight) × (distance from the additional weight to the pivot)
= 1 kg × 0.12 m (converting 12 cm to meters)
= 0.12 Nm
To find the total torque of the resistance, we add these two torques:
total torque of the resistance = torque1 + torque2
= 0.12 Nm + 0.12 Nm
= 0.24 Nm
The effort force required to support the arm and the weight is equal to the total torque divided by the distance from the force cord to the pivot:
effort force = total torque of the resistance / distance from the force cord to the pivot
= 0.24 Nm / 0.01 m (converting 1 cm to meters)
= 24 N
Therefore, an effort force of 24 Newtons will be required to support the arm and the additional weight.