Vector Decomposition and Dot Product yxθ r F θ r F r|| r±r||= rcos(θ ) r±= rsin(θ )θ r F F|| F±F||= Fcos(θ ) F±= Fsin(θ )Fonto rronto Fr • F = F • r = r F cos(θ) rdot Fr • F=r F cos(θ) = ±r||F= ±rF||²+ F r|| −F r||
Vector Decomposition and Cross Product yxθ r F θ r F r|| r±r||= rcos(θ ) r±= rsin(θ )|r ×F| = r±Fθ r F F|| F±F||= Fcos(θ ) F±= Fsin(θ )|r ×F| = rF±Fonto rronto F|r ×F|= r F sin(θ) rcross F²|r ×F| =r±F= rF±=r F sin(θ)
An Example of Cross Product Given r= 10 [m], F= 100 [N], and θ= 30º, calculate the cross product . !r×!F!r×!F=rFsinθ=(10)(100)sin(30°)=500 [mN]θ !F!ryxzr⊥F=[(10)sin(30°)](100)=500 [mN]rF⊥=(10)[(100)sin(30°)]=500 [mN]θ !F!rF⊥θ OR !r×!F=500 (−ˆz) = −500ˆz[mN] or !r×!F=0ˆx+0⌢y−500ˆz[mN] ²×θ !F!rr⊥θ Right-Hand Rule !r×!F
Definition: Torque, moment or moment of force, is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Work and Torque !τ=!r×!FDefinition: A force is said to do work when it acts on a body so that there is a displacement of the point of application in the direction of the force. W=!r⋅!FMore on these later…Note: Wis a scalar and "is a vector.
The quantity r×Fmeasures the “torque”; the ability of an applied force to cause rotation.
Scenario C: r×F= |r| |F| sin(θ) is maximized when and Fare perpendicular andwhen their magnitudes are maximized. Which scenario above gives the largest torque? rC
Rotation and Torque •The ability of a force to make something rotate depends on where you apply it and what direction you apply it in…. •This can be determined using the cross product If you want to spin this around its center, apply a force which acts at some spot on it and in some direction. FrAxis of rotation Definition:ris measured from the axis of rotation to the point where the force is applied.