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a particle of mass 3 kg is moving under the action of central force whose potential is U - 10/r3 j where r is from centre for what value energy will the path of the particle be a cirle radius 10m
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a particle of mass 3 kg is moving under the action of central force wh...
we know that force is given as the negative gradient of potential energy, so
F = -dU/dr
here,
U = 10r^3 J
so,
F = -d/dt( 10r3)
or F = -30r^2
thus,
F = centripetal force = mv^2/r = 30r2
so,
v^2 = 30 (r^3/ m)
here
m = 3kg
r = 10m
so,
v^2 = 30 x (103 / 3) = 10000
v = 100 m/s
now, kinetic energy
KE = (1/2)mv2 = (1/2)x3x1002
so,
KE = 15000 J
and potential energy
PE = 10r^3 = 10x103
so,
PE = 10000 J
thus, total energy
E = KE + PE = 15000 + 10000
or
E = 25000 J
and
angular momentum
L = mvr = 3 x 100 x 10
thus,
L = 3000 kg.m^2/s
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a particle of mass 3 kg is moving under the action of central force wh...
Introduction:
In this problem, we are given the potential energy function U of a central force acting on a particle. We need to determine the energy at which the path of the particle will be a circle with a radius of 10m.
Understanding the problem:
To understand the problem, let's first analyze the given potential energy function U. It is given as U - 10/r^3 j, where r represents the distance from the center. This potential energy function is dependent on the radial distance r and is in the form of a central force.
Conditions for a circular path:
For a particle to move in a circular path, the net force acting on the particle must be directed radially inwards towards the center. This can be achieved when the radial component of the force is equal to the centripetal force required for circular motion.
Deriving the force equation:
To find the force equation, we need to take the negative gradient of the potential energy function U. Since the given potential is in vector form, we need to take the derivative with respect to r to find the radial component of the force.
Taking the derivative of U with respect to r, we get:
F_r = -dU/dr = 30/r^4 j
Equating the radial force with centripetal force:
To find the energy at which the path of the particle will be a circle, we need to equate the radial force with the centripetal force.
Centripetal force = m*v^2 / r, where m is the mass of the particle, v is its velocity, and r is the radius of the circle.
Equating the radial force with the centripetal force, we get:
30/r^4 = m*v^2 / r
Simplifying the equation:
To simplify the equation, we can substitute the given mass value of 3kg and rearrange it to solve for v^2.
30/r^4 = (3*v^2) / r
10/r^4 = v^2 / r
Calculating the energy:
Now, we can calculate the energy at which the path of the particle will be a circle by considering the kinetic energy of the particle.
Kinetic energy = (1/2) * m * v^2
Substituting the mass value, we get:
Kinetic energy = (1/2) * 3 * (10/r^4)
Hence, the energy at which the path of the particle will be a circle with a radius of 10m is (1/2) * 3 * (10/r^4).
Conclusion:
In this problem, we analyzed the given potential energy function and derived the force equation. By equating the radial force with the centripetal force, we obtained an equation for the velocity of the particle. Finally, using the kinetic energy equation, we calculated the energy at which the path of the particle will be a circle.
In this problem, we are given the potential energy function U of a central force acting on a particle. We need to determine the energy at which the path of the particle will be a circle with a radius of 10m.
Understanding the problem:
To understand the problem, let's first analyze the given potential energy function U. It is given as U - 10/r^3 j, where r represents the distance from the center. This potential energy function is dependent on the radial distance r and is in the form of a central force.
Conditions for a circular path:
For a particle to move in a circular path, the net force acting on the particle must be directed radially inwards towards the center. This can be achieved when the radial component of the force is equal to the centripetal force required for circular motion.
Deriving the force equation:
To find the force equation, we need to take the negative gradient of the potential energy function U. Since the given potential is in vector form, we need to take the derivative with respect to r to find the radial component of the force.
Taking the derivative of U with respect to r, we get:
F_r = -dU/dr = 30/r^4 j
Equating the radial force with centripetal force:
To find the energy at which the path of the particle will be a circle, we need to equate the radial force with the centripetal force.
Centripetal force = m*v^2 / r, where m is the mass of the particle, v is its velocity, and r is the radius of the circle.
Equating the radial force with the centripetal force, we get:
30/r^4 = m*v^2 / r
Simplifying the equation:
To simplify the equation, we can substitute the given mass value of 3kg and rearrange it to solve for v^2.
30/r^4 = (3*v^2) / r
10/r^4 = v^2 / r
Calculating the energy:
Now, we can calculate the energy at which the path of the particle will be a circle by considering the kinetic energy of the particle.
Kinetic energy = (1/2) * m * v^2
Substituting the mass value, we get:
Kinetic energy = (1/2) * 3 * (10/r^4)
Hence, the energy at which the path of the particle will be a circle with a radius of 10m is (1/2) * 3 * (10/r^4).
Conclusion:
In this problem, we analyzed the given potential energy function and derived the force equation. By equating the radial force with the centripetal force, we obtained an equation for the velocity of the particle. Finally, using the kinetic energy equation, we calculated the energy at which the path of the particle will be a circle.
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a particle of mass 3 kg is moving under the action of central force whose potential is U - 10/r3 j where r is from centre for what value energy will the path of the particle be a cirle radius 10m
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a particle of mass 3 kg is moving under the action of central force whose potential is U - 10/r3 j where r is from centre for what value energy will the path of the particle be a cirle radius 10m for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about a particle of mass 3 kg is moving under the action of central force whose potential is U - 10/r3 j where r is from centre for what value energy will the path of the particle be a cirle radius 10m covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for a particle of mass 3 kg is moving under the action of central force whose potential is U - 10/r3 j where r is from centre for what value energy will the path of the particle be a cirle radius 10m.
a particle of mass 3 kg is moving under the action of central force whose potential is U - 10/r3 j where r is from centre for what value energy will the path of the particle be a cirle radius 10m for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about a particle of mass 3 kg is moving under the action of central force whose potential is U - 10/r3 j where r is from centre for what value energy will the path of the particle be a cirle radius 10m covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for a particle of mass 3 kg is moving under the action of central force whose potential is U - 10/r3 j where r is from centre for what value energy will the path of the particle be a cirle radius 10m.
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