Moment Arm (r)=Perpendicular Distance from the Line of Action to the Axis of Rotation

In mathematical terms, if the force is applied at an angle θ to the lever arm, the moment arm can be calculated as:

r=d×sin(θ)

Where:

  • r is the moment arm,
  • d is the perpendicular distance from the line of action to the axis of rotation,
  • θ is the angle between the force vector and the line connecting the force application point to the axis of rotation.

Case Study: Opening a Door

Imagine you are opening a door using a door handle. You exert a force on the handle to rotate the door about its hinges. In this scenario, we will calculate the moment arm and torque involved in opening the door.

Parameters:

  • Force Applied (F): 15 Newtons
  • Perpendicular Distance (d): 0.8 meters (distance from the door handle to the hinges)
  • Angle (θ): 90 degrees (assuming the force is applied perpendicular to the door)

Calculations: The moment arm (r) can be calculated using the formula r=d×sin⁡(θ)r=d×sin(θ), where dd is the perpendicular distance, and θ is the angle.

r=0.8 m×sin⁡(90∘)
r=0.8m×sin(90∘)
r=0.8 m×1
r=0.8 m

Now, the torque (τ) can be calculated using the formula τ=F×r, where F is the force applied, and r is the moment arm.

τ=15 N×0.8 m
τ=15N×0.8m
τ=12 Nm

Real-life Examples


Door Hinge:

  • Force Applied: The force exerted on the door.
  • Axis of Rotation: Door hinge.
  • Moment Arm: The distance from the hinge to the point where the force is applied.
  • Importance: A longer moment arm allows for easier opening or closing of the door.

Wrench Tightening a Bolt:

  • Force Applied: The force applied to turn the wrench.
  • Axis of Rotation: The center of the bolt.
  • Moment Arm: The distance from the center of the bolt to the point where the force is applied.
  • Importance: A longer moment arm increases the torque applied to the bolt.

Seesaw:

  • Force Applied: The downward force exerted by a person sitting on one end of the seesaw.
  • Axis of Rotation: The pivot point of the seesaw.
  • Moment Arm: The distance from the pivot point to the point where the force is applied.
  • Importance: The seesaw will rotate around the pivot point based on the relative lengths of the moment arms on each side.

Lever Systems in the Human Body:

  • Force Applied: Muscular forces.
  • Axis of Rotation: Joints (e.g., elbow joint, knee joint).
  • Moment Arm: The distance from the joint to the point where the muscle attaches.
  • Importance: The moment arm influences the effectiveness of muscle forces in producing joint movements.

Further Reading