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34kurt
3 years ago
11

A 5.0-g marble is released from rest in the deep end of a swimming pool. An underwater video reveals that its terminal speed in

the water is 0.30 m/s. (a) What is the acceleration of the marble at the instant it is released? (b) What is the acceleration of the marble when it has reached its terminal speed? (c) How long does it take the marble to reach half its terminal speed?
Physics
1 answer:
Temka [501]3 years ago
3 0

Answer:

a) a = -g = 9.8 m/s² , b) a = 0 m/s² and c)   t1 = 0.0213 s

Explanation:

a) At the moment the marble is released its velocity is zero, so it has no resistance force, the only force acting is its weight, so the acceleration is the acceleration of gravity

       a = -g = 9.8 m / s²

b) When the marble goes its terminal velocity all forces have been equalized, therefore, the sum of them is zero and consequently if acceleration is also zero

      a = 0 m / s²

c) We have to assume a specific type of resistive force, for liquid in general the resistive force is proportional to the speed of the body.

The expression of this situation is

         v = mg / b (1 -e^{-bt/m} )

For a very long time the exponential is zero, so the terminal velocity is

        v_{T} = mg / b

        b = mg /  v_{T}

        b = 5 10-3 9.8 / 0.3

        b = 0.163

We already have all the data to calculate the time for v = ½ v_{T}

        ½ v_{T} = v_{T} (1 -e^{-bt/m})

        ½ = 1- e (- 0.163 t1 / 5 10-3)

        e (-32.6 t1) = 1-0.5              (by  ln())

       -32.6 t1 = ln 0.5

       t1 = -1 / 32.6 (-0.693)

       t1 = 0.0213 s

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<u>Electrostatic Force</u>

Two point-charges q_1 and q_2 separated a distance d will exert a force on each other of a magnitude given by the Coulomb's formula

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The force between charges separated 2d is

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And the force between the charges A and D is

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Now, let's analyze each charge and the force applied to them by the others

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\displaystyle F_A=\frac{k\ Q^2}{d^2}(-1-\frac{1}{4}+\frac{1}{9})

\displaystyle F_A=-\frac{41}{36}F_1

Charge B. It receives force to the right from A and D and to the left from C

\displaystyle F_B=F_1-F_1+F_2=\frac{k\ Q^2}{d^2}-\frac{k\ Q^2}{d^2}+\frac{k\ Q^2}{4d^2}

\displaystyle F_B=\frac{1}{4}F_1

Charge C. It receives forces to the right from all charges.

\displaystyle F_C=F_2+F_1+F_1=\frac{k\ Q^2}{4d^2}+\frac{k\ Q^2}{d^2}+\frac{k\ Q^2}{d^2}

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Charge D. It receives forces to the left from all charges

\displaystyle F_D=-F_3-F_2-F_1=-\frac{k\ Q^2}{9d^2}-\frac{k\ Q^2}{4d^2}-\frac{k\ Q^2}{d^2}

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Comparing the magnitudes of each force is just a matter of computing the fractions

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a.

We can see the charge C experiences the greatest net force, and charge B receives the smallest net force

b.

The ratio of the greatest to the smallest net force is

\displaystyle \frac{\frac{9}{4}}{\frac{1}{4}}=9

The greatest force is 9 times the smallest net force

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