Pulling a Crate with Friction (Based on an OSU Problem Solving Exercise, Angle f
ID: 1396821 • Letter: P
Question
Pulling a Crate with Friction (Based on an OSU Problem Solving Exercise, Angle for Maximum acceleration. )
A wooden crate of mass M = 150.0 kg is to be pulled along a rough horizontal surface (floor) by an applied force
P = 580.0 N. The coefficient of friction (mu = ) between the crate and the floor is 0.380. Determine the optimum angle theta ( ) for pulling the crate; that is, the angle between the force P and the horizontal floor that produces the maximum acceleration. Recall the discussions and illustrations and equations we had on the board last week. For convenience, the acceleration equation is repeated here:
a = [ P*cos( ) – * ( M*g – P * sin ( ) ) ] / M
Explanation / Answer
I will right T for theta here.
As
a = [P cos(T) - u(Mg - P cos(T))]/M
Differentiating with respect to T, and setting the derivative to 0 (that's how we optimize),
da/dT = [-P sin(T) + u P cos(T)]/M = 0
Thus,
-P sin(T) + u P cos(T) = 0
-sin(T) + u cos(T) = 0
tan(T) = u
Thus,
T = arctan u
As u = 0.380,
T = acrtan 0.380
T = 20.81 degrees [ANSWER]
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