Principles of Physics: A Calculus-Based Text
Principles of Physics: A Calculus-Based Text
5th Edition
ISBN: 9781133104261
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 5, Problem 61P

(a)

To determine

Terminal velocity of water droplets at radii 10.0μm.

(a)

Expert Solution
Check Mark

Answer to Problem 61P

Terminal velocity of water droplets at radii 10.0μm is 0.0132m/s_.

Explanation of Solution

At the terminal velocity, the drag force is balanced by the force due to gravity.

    mg=arv+br2v2        (I)

Here, m is the mass, g is the acceleration due to gravity, r is the radius, v is then speed, and a,b are constants.

Write the expression for mass water drop.

    m=ρ43πr3        (II)

Here, ρ is the density.

Conclusion:

Substitute, 1000kg/m3, 105m for r in the equation (II), to find m.

    m=(1000kg/m3)43π(105m)3=4.186×1012kg

Substitute, 4.186×1012kg for m, 10.0μm for r, 3.10×104 for a, 0.870 for b, and 9.80m/s2 for g in the equation (I), and solve for v.

    (4.186×1012kg)(9.80m/s2)=(3.10×104)(10.0μm)v+(0.870)(10.0μm)2v24.11×1011=(3.10×109)v+(0.870×1010)v2

Assume that v is small, so neglect the second term in the right hand side.

    4.11×1011=(3.10×109)vv=4.11×10113.10×109=0.0132m/s

Therefore, terminal velocity of water droplets at radii 10.0μm is 0.0132m/s_.

(b)

To determine

Terminal velocity of water droplets at radii 100μm.

(b)

Expert Solution
Check Mark

Answer to Problem 61P

Terminal velocity of water droplets at radii 100μm is 1.03m/s_.

Explanation of Solution

Conclusion:

Substitute, 1000kg/m3, 100×106m for r in the equation (II), to find m.

    m=(1000kg/m3)43π(100×106m)3=4.186×109kg

Substitute, 4.186×109kg for m, 100×106m for r, 3.10×104 for a, 0.870 for b, and 9.80m/s2 for g in the equation (I), and solve for v.

    (4.186×109kg)(9.80m/s2)=(3.10×104)(100×106m)v+(0.870)(100μm)2v24.11×108=(3.10×108)v+(0.870×108)v2(0.870×108)v2+(3.10×108)v(3.10×108)v=0

Solve the above quadratic equation.

    v=3.10±(3.10)2+4(0.870)(4.11)2(0.870)=1.03m/s

Therefore, terminal velocity of water droplets at radii 100μm is 1.03m/s_.

(c)

To determine

Terminal velocity of water droplets at radii 1.00mm.

(c)

Expert Solution
Check Mark

Answer to Problem 61P

Terminal velocity of water droplets at radii 1.00mm is 6.87m/s_.

Explanation of Solution

Conclusion:

Substitute, 1000kg/m3, 1.00×103m1.00mm for r in the equation (II), to find m.

    m=(1000kg/m3)43π(1.00×103m)3=4.186×106kg

Substitute, 4.186×106kg for m, 1.00×103m1.00mm for r, 3.10×104 for a, 0.870 for b, and 9.80m/s2 for g in the equation (I), and solve for v.

    (4.186×106kg)(9.80m/s2)=(3.10×104)(1.00×103m)v+(0.870)(1.00×103m)2v24.11×105=(3.10×107)v+(0.870×106)v2

Assume that, v>1m/s, and neglect the first term in the above equation.

    4.11×105=(0.870×106)v2v=6.87m/s

Therefore, terminal velocity of water droplets at radii 1.00mm is 6.87m/s_.

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Chapter 5 Solutions

Principles of Physics: A Calculus-Based Text

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