4.10:
Moment of a Force About an Axis: Vector
Consider a cyclist pedaling a bicycle. When a force is exerted on the pedal, the crankshaft rotates such that the chain attached to the crankshaft exerts a force on the cogs of the wheel.
On establishing a coordinate system, the tangential force on the cogs is along the x-axis, while the radius of the cogs is the moment arm expressed in terms of the position vector.
Here, the moment of force about point O is along the y-direction perpendicular to both the force and position vector and is expressed as the cross product of the position and force vectors.
The cross-product can be solved by expressing the coefficient of components of force and position vector in the form of a determinant.
Expanding the determinant, the moment's component along the x-axis is expressed in terms of the y and z components of the force and position vector.
Similarly, expressing its other two components in terms of force and position vector components, the moment of a force can be expressed in cartesian vector form.
4.10:
Moment of a Force About an Axis: Vector
When a force is exerted on an object, it can cause that object to rotate about an axis. The moment of a force, also known as torque, measures the force's ability to cause that rotation. In the case of a cyclist pedaling a bicycle, the force exerted on the pedal causes the crankshaft to rotate, which in turn causes the wheel to spin. The moment of the force exerted on the pedal drives the wheel's rotation.
First, establish a coordinate system to understand how the moment of a force works. Consider the bicycle as an example. Establish the x-axis to be tangential to the wheel, and the y-axis to be perpendicular to the x-axis and pointing upwards. The z-axis can be pointing outwards from the wheel.
When the cyclist exerts a force on the pedal, the crankshaft rotates such that the chain attached to the crankshaft exerts a force on the axle of the wheel. The force exerted on the axle is then along the x-axis. The radius of the axle is the moment arm, which can be expressed in terms of the position vector.
The moment of force about point O is along the z-direction, perpendicular to both the force and position vector. It is expressed as the cross-product of the position and force vectors.
The cross-product can be solved by expressing the coefficient of components of force and position vector in the form of a determinant. The determinate can be expanded to solve for each component of the moment of force.
For example, the moment's component along the x-axis is expressed in terms of the y and z components of the force and position vector. Similarly, the other two components of the moment can be expressed in cartesian vector form.
Understanding the moment of force is important in many fields of science and engineering. It can help us calculate the forces on objects such as gears, pulleys, and levers and can aid in designing machines that convert forces into movement. In the bicycle example, understanding the moment of force can help us optimize the design of the crankshaft and chain to maximize the force exerted on the wheel.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics ‒ Statics and Dynamics. Hoboken, New Jersey: Pearson Prentice Hall. pp 146
- Beer, F.P.; Johnston, E.R.; Mazurek, D.F; Cromwell, P.J. and Self, B.P.(2019). Vector Mechanics for Engineers ‒ Statics and Dynamics. New York: McGraw-Hill. pp 107 – 109