Tug War I
Classical Mechanics
Level
3
Arnie and Wimpy are playing tug-of-war. Arnie wins, pulling Wimpy into the mud puddle. Why did Arnie win?
Arnie exerted a greater horizontal component of force on the rope than Wimpy exerted.
All of the above.
Arnie exerted a greater horizontal component of force on the ground than Wimpy exerted.
The rope exerted a greater horizontal component of force on Wimpy than it exerted on Arnie.
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At the start everyone is at rest. For someone to win, there must be an acceleration from rest to initiate motion in one direction. This acceleration need not be large. During the acceleration there must be a net horizontal force component on each person in the direction of the acceleration.
(a) and (b) are wrong because the rope has relatively small mass, so the tension forces at its ends are nearly the same size (F =ma where m is essentially zero). Certainly any small difference in tension between the two ends is nowhere near great enough to account for the outcome. The rope's mass is very small compared to the masses of Arnie and Wimpy, so even when it accelerates (and so do they), the net force on the rope, F = ma, is negligibly small compared to the net forces on Arnie and Wimpy. In fact, most of this game involves small accelerations.
(c) is correct. The net horizontal force component on Arnie that allows him to accelerate with respect to the ground is the force of the ground on his feet minus the force the rope exerts on him, the former being larger. The person with the better shoes, greater coefficient of friction at the ground underfoot, and foot control, wins. Wimpy also accelerates, because the force of the rope on him is larger than the force of the ground on his feet. The player wins who can maintain the largest horizontal force between his feet and the ground.