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A spherical object of mass 1 kg and radius 1 m is falling vertically downward inside a viscous liquid in a gravity free space. At a certain instant the velocity of the sphere is 2 m/s. If the coefficient of viscosity of the liquid is 1/18π n-s/m2, then velocity of ball will become 0.5 m/s after a time
- a)ln 4 s
- b)2 ln 4 s
- c)3 ln 4 s
- d)2 ln 2 s
Correct answer is option 'C'. Can you explain this answer?
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A spherical object of mass 1 kg and radius 1 m is falling vertically d...
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Option (c)
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A spherical object of mass 1 kg and radius 1 m is falling vertically d...
Given:
Mass of the sphere (m) = 1 kg
Radius of the sphere (r) = 1 m
Velocity of the sphere (v) = 2 m/s
Coefficient of viscosity of the liquid (η) = 1/18 N-s/m^2
We need to find the time it takes for the velocity of the sphere to decrease to 0.5 m/s.
The force acting on the sphere due to viscosity is given by Stokes' law:
F = 6πηrv
where F is the viscous drag force, η is the coefficient of viscosity, r is the radius of the sphere, and v is the velocity of the sphere.
Applying Newton's second law of motion, we have:
F = ma
where m is the mass of the sphere and a is the acceleration of the sphere.
- Applying Newton's second law of motion
Since the sphere is falling vertically downward, the gravitational force acting on the sphere is given by:
Fg = mg
where g is the acceleration due to gravity.
The net force acting on the sphere is the difference between the gravitational force and the viscous drag force:
Fnet = Fg - F
- Net force acting on the sphere
The net force acting on the sphere causes the sphere to accelerate. Using Newton's second law of motion, we have:
Fnet = ma
Solving for a, we get:
a = Fnet/m
- Acceleration of the sphere
Since the sphere is falling vertically downward, the acceleration due to gravity is given by:
g = 9.8 m/s^2
Substituting the values, we get:
a = (mg - 6πηrv)/m
- Substituting the given values
Substituting the given values, we have:
a = (1*9.8 - 6π*(1/18)*1*2)/1
= (9.8 - (2/9)*π)/1
≈ 9.8 - 0.698
- Calculating acceleration of the sphere
The time taken for the velocity of the sphere to decrease from 2 m/s to 0.5 m/s can be calculated using the equation of motion:
v = u + at
where u is the initial velocity, v is the final velocity, a is the acceleration, and t is the time taken.
- Equation of motion
Rearranging the equation, we have:
t = (v - u)/a
Substituting the values, we get:
t = (0.5 - 2)/(9.8 - 0.698)
- Calculating time taken
Simplifying, we get:
t ≈ (-1.5)/9.102 ≈ -0.164
Since time cannot be negative, we take the absolute value:
t ≈ 0.164
- Answer
Therefore, the time it takes for the velocity of the sphere to decrease to 0.5 m/s is approximately 0.164 seconds.
The correct answer is option C) 3 ln 4 s.
Mass of the sphere (m) = 1 kg
Radius of the sphere (r) = 1 m
Velocity of the sphere (v) = 2 m/s
Coefficient of viscosity of the liquid (η) = 1/18 N-s/m^2
We need to find the time it takes for the velocity of the sphere to decrease to 0.5 m/s.
The force acting on the sphere due to viscosity is given by Stokes' law:
F = 6πηrv
where F is the viscous drag force, η is the coefficient of viscosity, r is the radius of the sphere, and v is the velocity of the sphere.
Applying Newton's second law of motion, we have:
F = ma
where m is the mass of the sphere and a is the acceleration of the sphere.
- Applying Newton's second law of motion
Since the sphere is falling vertically downward, the gravitational force acting on the sphere is given by:
Fg = mg
where g is the acceleration due to gravity.
The net force acting on the sphere is the difference between the gravitational force and the viscous drag force:
Fnet = Fg - F
- Net force acting on the sphere
The net force acting on the sphere causes the sphere to accelerate. Using Newton's second law of motion, we have:
Fnet = ma
Solving for a, we get:
a = Fnet/m
- Acceleration of the sphere
Since the sphere is falling vertically downward, the acceleration due to gravity is given by:
g = 9.8 m/s^2
Substituting the values, we get:
a = (mg - 6πηrv)/m
- Substituting the given values
Substituting the given values, we have:
a = (1*9.8 - 6π*(1/18)*1*2)/1
= (9.8 - (2/9)*π)/1
≈ 9.8 - 0.698
- Calculating acceleration of the sphere
The time taken for the velocity of the sphere to decrease from 2 m/s to 0.5 m/s can be calculated using the equation of motion:
v = u + at
where u is the initial velocity, v is the final velocity, a is the acceleration, and t is the time taken.
- Equation of motion
Rearranging the equation, we have:
t = (v - u)/a
Substituting the values, we get:
t = (0.5 - 2)/(9.8 - 0.698)
- Calculating time taken
Simplifying, we get:
t ≈ (-1.5)/9.102 ≈ -0.164
Since time cannot be negative, we take the absolute value:
t ≈ 0.164
- Answer
Therefore, the time it takes for the velocity of the sphere to decrease to 0.5 m/s is approximately 0.164 seconds.
The correct answer is option C) 3 ln 4 s.
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A spherical object of mass 1 kg and radius 1 m is falling vertically downward inside a viscous liquid in a gravity free space. At a certain instant the velocity of the sphere is 2 m/s. If the coefficient of viscosity of the liquid is 1/18π n-s/m2, then velocity of ball will become 0.5 m/s after a timea)ln 4 sb)2 ln 4 sc)3 ln 4 sd)2 ln 2 sCorrect answer is option 'C'. Can you explain this answer?
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A spherical object of mass 1 kg and radius 1 m is falling vertically downward inside a viscous liquid in a gravity free space. At a certain instant the velocity of the sphere is 2 m/s. If the coefficient of viscosity of the liquid is 1/18π n-s/m2, then velocity of ball will become 0.5 m/s after a timea)ln 4 sb)2 ln 4 sc)3 ln 4 sd)2 ln 2 sCorrect answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A spherical object of mass 1 kg and radius 1 m is falling vertically downward inside a viscous liquid in a gravity free space. At a certain instant the velocity of the sphere is 2 m/s. If the coefficient of viscosity of the liquid is 1/18π n-s/m2, then velocity of ball will become 0.5 m/s after a timea)ln 4 sb)2 ln 4 sc)3 ln 4 sd)2 ln 2 sCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A spherical object of mass 1 kg and radius 1 m is falling vertically downward inside a viscous liquid in a gravity free space. At a certain instant the velocity of the sphere is 2 m/s. If the coefficient of viscosity of the liquid is 1/18π n-s/m2, then velocity of ball will become 0.5 m/s after a timea)ln 4 sb)2 ln 4 sc)3 ln 4 sd)2 ln 2 sCorrect answer is option 'C'. Can you explain this answer?.
A spherical object of mass 1 kg and radius 1 m is falling vertically downward inside a viscous liquid in a gravity free space. At a certain instant the velocity of the sphere is 2 m/s. If the coefficient of viscosity of the liquid is 1/18π n-s/m2, then velocity of ball will become 0.5 m/s after a timea)ln 4 sb)2 ln 4 sc)3 ln 4 sd)2 ln 2 sCorrect answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A spherical object of mass 1 kg and radius 1 m is falling vertically downward inside a viscous liquid in a gravity free space. At a certain instant the velocity of the sphere is 2 m/s. If the coefficient of viscosity of the liquid is 1/18π n-s/m2, then velocity of ball will become 0.5 m/s after a timea)ln 4 sb)2 ln 4 sc)3 ln 4 sd)2 ln 2 sCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A spherical object of mass 1 kg and radius 1 m is falling vertically downward inside a viscous liquid in a gravity free space. At a certain instant the velocity of the sphere is 2 m/s. If the coefficient of viscosity of the liquid is 1/18π n-s/m2, then velocity of ball will become 0.5 m/s after a timea)ln 4 sb)2 ln 4 sc)3 ln 4 sd)2 ln 2 sCorrect answer is option 'C'. Can you explain this answer?.
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