Equivalence Claims r F r F and r F r F Example Given r 10 m F 50 N and \u03b8 60\u00ba

Equivalence claims r f r f and r f r f example given

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Vector Decomposition and Dot Product y x θ r F θ r F r || r ± r || = r cos( θ ) r ± = r sin( θ ) θ r F F || F ± F || = F cos( θ ) F ± = F sin( θ ) F onto r r onto F r F = F r = r F cos( θ ) r dot F r F = r F cos ( θ ) = ± r || F = ± r F || ² + F r || F r ||
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Vector Decomposition and Cross Product y x θ r F θ r F r || r ± r || = r cos( θ ) r ± = r sin( θ ) |r × F | = r ± F θ r F F || F ± F || = F cos( θ ) F ± = F sin( θ ) |r × F | = r F ± F onto r r onto F | r × F | = r F sin( θ ) r cross F ² |r × F | = r ± F = r F ± = 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 = rF sin θ = (10)(100)sin(30 ° ) = 500 [mN] θ ! F ! r y x z r F = [(10)sin(30 ° )](100) = 500 [mN] rF = (10)[(100)sin(30 ° )] = 500 [mN] θ ! F ! r F θ OR ! r × ! F = 500 ( ˆ z ) = 500 ˆ z [mN] or ! r × ! F = 0 ˆ x +0 y 500 ˆ z [mN] ² × θ ! F ! r r 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 × ! F Definition: 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 ! F More on these later Note: W is 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? r C
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 . F r Axis of rotation Definition: r is measured from the axis of rotation to the point where the force is applied.