4.9:
Moment of a Force About an Axis: Scalar
Consider a cyclist riding a bicycle and applying force on the pedal to turn the wheel. When a force is exerted on the pedal, the attached chain exerts a force on the wheel's cogs. The wheel rotates around its axle, so the axis of rotation passes through it.
The moment of a force along the axis measures how much the force causes an object to rotate around a specific point or axis.
In this case, the wheel rotates about the y-axis in the x–z plane. Here, the moment arm or the perpendicular distance between the axis of rotation and the line of action of the force is the radius of the cogs.
So the moment of a force about the y-axis is equal to Frsinθ.
Since the force is tangential, the angle between the force vector and the radius is 90 degrees.
In a bicycle, the moment of force can be multiplied by the angular velocity of the axle to obtain the power generated by the wheel rotation.
4.9:
Moment of a Force About an Axis: Scalar
The moment of a force about an axis is a crucial concept in mechanics that helps determine an object's rotational motion around a specific point or axis. The moment of force can be calculated using scalar analysis, which involves considering the perpendicular distance between the axis of rotation and the line of action of the force or simply the moment arm.
To better understand the concept of moment of force, consider the example of a cyclist riding a bicycle. When the cyclist applies force on the pedal, the attached chain exerts a force on the wheel's axle, causing it to rotate. Here, the axis of rotation passes through the wheel's axle and the moment arm is the radius of the axle. In this case, the moment of the force about the axis of rotation is given by:
where, F is the applied force, r is the radius of the axle, and θ is the angle between the force vector and the radius. Since the force is tangential, the angle between the force vector and the radius is 90 degrees.
In a bicycle, the moment of force can be multiplied by the angular velocity of the axle to obtain the power generated by the wheel rotation. The moment of force is essential in designing various structures and machines, including bicycles, automobiles, and aircraft, to ensure their stability and safety.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics ‒ Statics and Dynamics. Hoboken, New Jersey: Pearson Prentice Hall. pp 145.