A body of mass 2 kg is moving under the influence of a central force whose potential energy is given by U(r)= 2r3 Joule, If the body is moving in a circular orbit of 5m, its energy will be______ JCorrect answer is '625'. Can you explain this answer?

Question

Answer

Given information:

Mass of the body, m = 2 kg

Potential energy, U(r) = 2r^3 J

Radius of circular orbit, r = 5 m

Calculating the kinetic energy:

In a circular orbit, the centripetal force is provided by the central force. The centripetal force can be given by:

F = mω^2r

where m is the mass of the body, ω is the angular velocity, and r is the radius of the circular orbit.

The angular velocity ω can be related to the potential energy U(r) using the equation:

ω^2 = -(1/m) dU(r)/dr

Substituting the given potential energy U(r) = 2r^3 J:

ω^2 = -(1/m) d(2r^3)/dr

ω^2 = -(1/m) * 6r^2

ω^2 = -3r^2/m

The kinetic energy of the body can be calculated using the equation:

K = (1/2) mω^2 r^2

Substituting the value of ω^2:

K = (1/2) m * (-3r^2/m) * r^2

K = (-3/2) r^4

Calculating the total energy:

The total energy, E, is the sum of the kinetic energy, K, and the potential energy, U(r):

E = K + U(r)

E = (-3/2) r^4 + 2r^3

E = 2r^3 - (3/2) r^4

Substituting the value of r = 5:

E = 2(5^3) - (3/2) (5^4)

E = 500 - (3/2) (625)

E = 500 - 937.5

E = -437.5 J

Conclusion:

The energy of the body moving in a circular orbit of radius 5 m is -437.5 J.

Answer

By the effective central potential

for the circular orbit

total Energy =

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