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You have been asked to predict how the pressure inside a basketball changes as a player dribbles the ball. For a first model, assume the basketball can be modeled as a sphere of constant radius r containing pressurized air and the air can be modeled as an ideal gas. ... during a dribble, a section of the sphere deforms and is compressed a distance h by the floor without wrinkling or changing r. The result is a new spherical shape missing a segment (see figure). This missing segment is called a spherical cap and has a volume cap = (1/3) [h² (3r – h)]. Before Dribble Pinitial 150 kPa T = 27°C 2r r = 12 cm During Dribble Pdribble = ???? T = 27°C a) Calculate the mass of air, in kg, inside the basketball before the dribble. b) If the sphere is compressed a distance h = 3 cm during the dribble, determine the ratio P dribble/Pinitial. If necessary, assume the temperature of the air does not change and the mass inside the basketball is constant. c) If the mass of air inside the basketball, m = pV, is a constant, show that dp/dt = - − (p/¥)(d\/dt) where p is the uniform density of the air inside the basketball and Vis the volume of the air. Note p and both depend on time.

You have been asked to predict how the pressure inside a basketball changes as a player dribbles the ball. For a first model, assume the basketball can be modeled as a sphere of constant radius r containing pressurized air and the air can be modeled as an ideal gas. ... during a dribble, a section of the sphere deforms and is compressed a distance h by the floor without wrinkling or changing r. The result is a new spherical shape missing a segment (see figure). This missing segment is called a spherical cap and has a volume cap = (1/3) [h² (3r – h)]. Before Dribble Pinitial 150 kPa T = 27°C 2r r = 12 cm During Dribble Pdribble = ???? T = 27°C a) Calculate the mass of air, in kg, inside the basketball before the dribble. b) If the sphere is compressed a distance h = 3 cm during the dribble, determine the ratio P dribble/Pinitial. If necessary, assume the temperature of the air does not change and the mass inside the basketball is constant. c) If the mass of air inside the basketball, m = pV, is a constant, show that dp/dt = - − (p/¥)(d\/dt) where p is the uniform density of the air inside the basketball and Vis the volume of the air. Note p and both depend on time.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Problem 6.3 You have been asked to predict how the pressure inside a basketball changes as a player dribbles the ball. For a first model, assume the basketball can be modeled as a sphere of constant radius r containing pressurized air and the air can be modeled as an ideal gas. during a dribble, a section of the sphere deforms and is compressed a distance h by the floor without wrinkling or changing r. The result is a new spherical shape missing a segment (see figure). This missing segment is called a spherical cap and has a volume cap = (1/3) [h² (3r – h)]. Before Dribble Pinitial 150 kPa T = 27°C 2r - r = 12 cm During Dribble Pdribble = ???? T = 27°C a) Calculate the mass of air, in kg, inside the basketball before the dribble. b) If the sphere is compressed a distance h = 3 cm during the dribble, determine the ratio Pdribble/Pinitial. If necessary, assume the temperature of the air does not change and the mass inside the basketball is constant. h c) If the mass of air inside the basketball, m = p, is a constant, show that dp/dt = (p/¥)(d\/dt) where p is the uniform density of the air inside the basketball and Vis the volume of the air. Note p and both depend on time. cm. d) Write an expression for the volume inside the basketball in terms of r and h that would be valid at any time during the dribble. (Your answer will only contain symbols and pure numbers.) e) Using your result from Part (d), calculate dv/dt in terms of dh/dt, h, and r, and then determine the numerical value for K in the equation, dv/dt = K (dh/dt) when h = 3 Ans: a) 12.5 g ≤ m ≤ 12.75 g b) 0.95 ≤ Paribble /Pinitial ≤ 1.10; e) /180 cm² | <K</200 cm²|
Transcribed Image Text:Problem 6.3 You have been asked to predict how the pressure inside a basketball changes as a player dribbles the ball. For a first model, assume the basketball can be modeled as a sphere of constant radius r containing pressurized air and the air can be modeled as an ideal gas. during a dribble, a section of the sphere deforms and is compressed a distance h by the floor without wrinkling or changing r. The result is a new spherical shape missing a segment (see figure). This missing segment is called a spherical cap and has a volume cap = (1/3) [h² (3r – h)]. Before Dribble Pinitial 150 kPa T = 27°C 2r - r = 12 cm During Dribble Pdribble = ???? T = 27°C a) Calculate the mass of air, in kg, inside the basketball before the dribble. b) If the sphere is compressed a distance h = 3 cm during the dribble, determine the ratio Pdribble/Pinitial. If necessary, assume the temperature of the air does not change and the mass inside the basketball is constant. h c) If the mass of air inside the basketball, m = p, is a constant, show that dp/dt = (p/¥)(d\/dt) where p is the uniform density of the air inside the basketball and Vis the volume of the air. Note p and both depend on time. cm. d) Write an expression for the volume inside the basketball in terms of r and h that would be valid at any time during the dribble. (Your answer will only contain symbols and pure numbers.) e) Using your result from Part (d), calculate dv/dt in terms of dh/dt, h, and r, and then determine the numerical value for K in the equation, dv/dt = K (dh/dt) when h = 3 Ans: a) 12.5 g ≤ m ≤ 12.75 g b) 0.95 ≤ Paribble /Pinitial ≤ 1.10; e) /180 cm² | <K</200 cm²|
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