A father (mF = 80 kg) and his daughter (mD= 50 kg) stand on a flat frozen lake o
ID: 1352906 • Letter: A
Question
A father (mF = 80 kg) and his daughter (mD= 50 kg) stand on a flat frozen lake of negligible friction. They hold a 14 m long rope stretched between them. The father and daughter then pull the rope to bring them together. If the father is initially standing at the origin, how far from the origin will they meet?
A father (mF = 80 kg) and his daughter (mD= 50 kg) stand on a flat frozen lake of negligible friction. They hold a 14 m long rope stretched between them. The father and daughter then pull the rope to bring them together. If the father is initially standing at the origin, how far from the origin will they meet?
A proposed space station includes living quarters in a circular ring 56.7 m in diameter. At what angular speed should the ring rotate so the occupants feel 0.624 g where g is the gravitational acceleration on the surface of the Earth?
Explanation / Answer
The tension in the rope is the same at both ends of the rope. The sum of the distances that the father and daughter move = 14 m.
And the times of motion of the father and daughter are the same.
The tension is the force which is causing the father and daughter to accelerate toward each other.
Since the daughter has less mass than the father, the daughter will have a greater acceleration than the father.
So, the daughter will move farther, than the father.
Force = mass * acceleration
For daughter, F = mD * aD
For father, F = mF * aF
mD * aD = mF * aF
50 * aD = 80 * aF
aD = 80/50 * aF
The daughter’s acceleration is approximately 1.8 times the acceleration of the father.
distance = vi * t + ½ * acceleration * time^2
The initial velocity for both = 0 m/s
d = ½ * a * t^2
Time = (2 * a * d)^0.5
For daughter, t = (2 * aD * dD)^0.5
For father, t = (2 * aF * dF)^0.5
The times are equal.
(2 * aD * dD)^0.5 = (2 * aF * dF)^0.5
2 * aD * dD = 2 * aF * dF
aD * dD = aF * dF
Substitute 80/50 * aF for aD
80/50 * aF * dD = aF * dF
80/50 * dD = dF
The sum of the distances = 14 m
dD + 80/50 * dD = 14
dD = 14 ÷ (1+ 80/50) = 5.385 m
The daughter moved 5.385 m, so the father moved 14 – 5.384 = 8.616 m
dD/ dF = 8.616/5.385 = 1.6
The daughter’s distance is approximately 1.6 times the acceleration of the father’s distance,
because the daughter’s acceleration is approximately 1.6 times the acceleration of the father;
because the father’s mass is approximately 1.6 times the daughter’s mass.
All this work and the answer was just an inverse ratio!!
The distance is inversely proportional to the mass!
mD/mF = 50/80, dD/ dF = 80/50
The daughter moves 80/50 times the distance that the father moves.
The work above is the proof of the inverse relationship!
The proof is always longer than the equation!
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