Two people,one twice as massive as the other,attempt a tug-of-war with 12 m of rope on frictionless ice.After a brief time,they meet.The heavier person slides a distance of
A)3 m.
B)4 m.
C)5 m.
D)6 m.

Question

Quizplus · Accepted Answer

Since the rope is on frictionless ice, the tension in the rope will be constant throughout the tug-of-war. Let the tension be T. Since the two people meet after a brief time, the total distance they pulled the rope is equal to 12 m. Let the distance moved by the lighter person be x. Then the distance moved by the heavier person is (12-x).

By Newton's second law, F=ma, where F is the force applied, m is the mass of the person, and a is the acceleration. Since the tension in the rope is equal to the force applied by each person, we have T=ma. Rearranging this equation, we get a=T/m.

Let M be the mass of the heavier person and m be the mass of the lighter person. Since the heavier person is twice as massive as the lighter person, we have M=2m.

When the two people meet, the net force on each person is zero, so the accelerations are equal in magnitude but opposite in direction. That is, a = (T/m) = -(T/M). Substituting M=2m, we get a = -(T/2m).

We can now write expressions for the distances moved by each person:

x = (1/2)at^2 = (1/2)(T/m)t^2
12-x = (1/2)at^2 = (1/2)(T/2m)t^2

where t is the time elapsed. Since the two people meet after a brief time, we can assume that the times taken by each person are equal, so we have:

(1/2)(T/m)t^2 = (1/2)(T/2m)t^2
t^2 = (m/2M) t^2
t^2 = 1/3

Finally, we can substitute t^2=1/3 into the expressions for x and (12-x) to get:

x = (1/2)(T/m)(1/3) = T/6m
12-x = (1/2)(T/2m)(1/3) = T/12m

The heavier person slides a distance of (12-x) = T/12m. Since M=2m, we have T = Mg = 2mg, where g is the acceleration due to gravity. Substituting these values, we get:

(12-x) = T/12m = (2mg)/(12m) = g/6
So the heavier person slides a distance of 12-x = 12 - (g/6) = 4 m.

Therefore, the best choice is B) 4 m.

Two People,one Twice as Massive as the Other,attempt a Tug-Of-War