4.4
Moment of a Force: Vector Formulation
The moment of force refers to the measure of the rotational tendency of a force. It occurs when a force is applied in such a way that it produces a tw…
Consider a force applied perpendicular to the wrench that rotates the bolt about point O.
The position vector represents the distance of the point of force application to the axis of rotation.
The moment of a force is created perpendicular to the plane containing the point O, position vector, and force vector. It is directed along the axis of rotation.
The vector formulation of the moment of force is the cross-product of the position and force vectors.
The magnitude of the moment of force is Frsin-theta, where the moment arm is rsin-theta. So, the magnitude of the moment is maximum when the position vector and force are perpendicular to each other.
The right-hand rule determines its direction, which states that if fingers curl in the direction of the object's rotation, the thumb shows the direction of the moment.
If the force is applied at two different points but along the same line of action, it creates the same moment of force. This property is called the principle of transmissibility of force.
View the full transcript and gain access to JoVE Core videos
Q1: What is the vector formulation of the moment of a force?
The vector formulation of the moment of a force is the cross-product of the position vector and force vector. The cross-product results in a vector perpendicular to both the position and force vectors. This formulation is expressed as MO = r × F, where r is the position vector from the axis of rotation to the point of force application, and F is the force vector.
Q2: How do you calculate the magnitude of a moment of force?
The magnitude of the moment of force is calculated using the formula M = Fr sin θ, where F is the force magnitude, r is the position vector magnitude, and θ is the angle between them. The moment arm, r sin θ, represents the perpendicular distance from the line of action of the force to the axis of rotation. Magnitude is maximum when the position vector and force are perpendicular.
Q3: What does the right-hand rule tell you about moment direction?
The right-hand rule determines the direction of the moment vector. If your fingers curl in the direction of the object's rotation, your thumb points in the direction of the moment. This means the moment direction is perpendicular to the plane containing both the position and force vectors, aligned with the axis of rotation.
Q4: What is the principle of transmissibility of force?
The principle of transmissibility states that if a force is applied at two different points but along the same line of action, it creates the same moment of force about a given axis. This principle is fundamental in mechanics because it allows forces to be moved along their line of action without changing the rotational effect they produce.
Q5: Why is the moment of force a vector quantity rather than a scalar?
The moment of force is a vector quantity because it has both magnitude and direction. The magnitude represents the rotational tendency, while the direction indicates the axis and sense of rotation. Since the moment is defined as a cross-product of two vectors, the result is inherently a vector perpendicular to both input vectors.
Q6: How does the moment arm relate to the moment of a force?
The moment arm is the perpendicular distance from the line of action of the force to the axis of rotation, expressed as r sin θ. It directly determines the magnitude of the moment: larger moment arms produce larger moments for the same force. When the force is perpendicular to the position vector, the moment arm equals the full position vector length, maximizing rotational effect.
Q7: When is the moment of a force maximum?
The moment of a force is maximum when the position vector and force vector are perpendicular to each other, meaning θ = 90°. At this angle, sin θ = 1, so the magnitude reaches its peak value of M = Fr. Any other angle between the vectors produces a smaller moment magnitude.