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Moment and Torque Outline ! ! Ozkaya and Nordin, Ch. 3 (p. 31-46) ! ! ! ! Moments cause twisting or bending ! ! ! Moment is force times distance effect of a force F on a rigid body depends on where it is applied torque (T) or moment (M): measure of the tendency of a force to cause rotation or bending about a specific axis units of N·m M F definition of moment and moment arm conventions for defining moment direction addition of moments, moments due to force components moments due to external forces moments due to muscles translation of forces and equivalent systems ! the moment that a force produces about a fixed point O is equal to the force times the perpendicular distance from O to the line of action of the force d2 d1 F = mg r v MO = Fd ! d ! = "moment arm" r F (M ) O 2 T O F1 = 100 N F2 = 100 N r > ( MO )1 Moment arm is perpendicular to line of action of force Make sure you calculate moment based on the moment arm, not the distance from the point of force application to the axis of rotation. Moment is a Vector ! F1 d1 F2 ! d3 F3 P ! a moment has both magnitude and direction in static equilibrium problems, there is no strict convention for defining positive and negative moments we will usually refer to a CCW moments as positive, and a CW moments as negative Right Hand Rule ! ! moments can be represented in diagrams one of two ways the direction of MO is ! ! ! ! ! perpendicular to the plane formed by F and O determined by the “right hand rule” right hand rule: identify the direction that the moment tends to cause the body to rotate about point O orient your right hand so this direction is illustrated by the act of curling your fingers into increased flexion your thumb will be pointing in the direction of the moment Moment addition ! representation that shows the direction of MO r F ! Moments about a given axis add algebraically Be careful to account for the sign (i.e., direction) of each moment r M + MO = !2 " W2 ! 4 " W1 Moment calculations can be simplified by using force components ! It is often convenient to resolve a force F into its x, y, and z components in order to calculate the moment it generates at a specific location. External Forces Create Moments The effect of external loads on producing human body movements, and generating internal forces and stresses, depends on the moments they create. Consider, for example, how the angle ! influencces the feasibility or effort involved in the bench press exercise. + MB = Fy (x A ! x B ) ! Fx (yA ! yB ) Muscles Create Moments The ability of muscles to stabilize the body under external loads, and produce human body movements, is due their ability to create moments (or torques) about joints. If the line of action of a muscle force passes through a joint axis of rotation, it cannot stabilize or rotate that joint. Muscles have small moment arms Compared to external loads, muscles have small moment arms with respect to joint they span. Advantage: given the limited deformation that muscles can undergo, their close proximity to joints allows for large ranges of joint rotation Disadvantage: high muscle forces (and therefore joint forces) are required to balance external loads Forces can be moved by replacing them with a force and a moment Elbow angle effects moment arm of elbow flexor muscles • Moment Arm (mm) 100 Brachialis 75 Biceps • A force F applied to a rigid body at a location A can be translated to another location O on the body, if a moment (or couple) MO is also added at O. The location of O will affect the magnitude of MO. Brachioradialis 50 2D 25 0 50 100 3D 150 Elbow Angle (degrees) Multiple forces can be replaced with a force and a moment • • Multiple forces applied to a rigid body can be replaced by a single resultant force and a single moment, acting at any location on the body. The location will affect the resultant moment, but not the resultant force. Review Questions ! ! ! ! ! ! Multiple external loads Equivalent resultant load and moment Equivalent single load How is a moment defined (mathematically)? How is a moment arm defined? How is the right hand rule used to determine the direction of a moment? How do moments add? How can several forces be replaced with a single moment and force? How does elbow rotation affect the moment arm of elbow flexor muscles? Does this explain why elbow strength varies with elbow angle? Tutorial problems: 1. Determine the magnitude of Mx, My, and Mz at the origin O, in terms of force components Fx, Fy, and Fz, and distances xA, yA, and zA. 2. Determine moments at the elbow (point B) and shoulder (point A) due to the vertical load F.