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Work Power and Energy question

2018 · 16 Apr · Shift 1 · Q66
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Work Power and Energy question

2018 · 16 Apr · Shift 1 · Q66

JEE MainPhysicsWork Power and EnergyMCQ+4 / −1
Two particles of the same mass m are moving in circular orbits because of force, given by F(r)=−16r−r3F\left( r \right) = {{ - 16} \over r} - {r^3}F(r)=r−16​−r3 The first particle is at a distance r = 1, and the second, at r = 4. The best estimate for the ratio of kinetic energies of the first and the second particle is closest to :
  1. A
    6×10−26 \times {10^{ - 2}}6×10−2
  2. B
    3×10−33 \times {10^{ - 3}}3×10−3
  3. C
    10−1{10^{ - 1}}10−1
  4. D
    6×1026 \times {10^{ 2}}6×102
View written solutionFree

Correct answer: A

  1. Condition for circular motion

For a particle of mass mmm moving in a circular orbit of radius rrr under a central force F(r)F(r)F(r), the required centripetal force is

mv2r=−F(r)\frac{mv^2}{r} = -F(r)rmv2​=−F(r)

since the given force is attractive (negative sign indicates inward direction).

Given:

F(r)=−16r−r3F(r)= -\frac{16}{r} - r^3F(r)=−r16​−r3

So its magnitude is

∣F(r)∣=16r+r3|F(r)| = \frac{16}{r}+r^3∣F(r)∣=r16​+r3

Hence,

mv2r=16r+r3\frac{mv^2}{r} = \frac{16}{r}+r^3rmv2​=r16​+r3

Multiplying by rrr:

mv2=16+r4mv^2 = 16 + r^4mv2=16+r4
  1. Kinetic energy at radius rrr

Kinetic energy is

K=12mv2=12(16+r4)K = \frac12 mv^2 = \frac12(16+r^4)K=21​mv2=21​(16+r4)

So for any circular orbit,

K(r)=16+r42K(r)=\frac{16+r^4}{2}K(r)=216+r4​
  1. For the first particle: r=1r=1r=1
K1=16+142=172K_1 = \frac{16+1^4}{2} = \frac{17}{2}K1​=216+14​=217​
  1. For the second particle: r=4r=4r=4
K2=16+442=16+2562=2722=136K_2 = \frac{16+4^4}{2} = \frac{16+256}{2} = \frac{272}{2}=136K2​=216+44​=216+256​=2272​=136
  1. Ratio of kinetic energies
K1K2=17/2136=17272=116\frac{K_1}{K_2} = \frac{17/2}{136} = \frac{17}{272} = \frac1{16}K2​K1​​=13617/2​=27217​=161​

Now,

116=0.0625=6.25×10−2\frac1{16}=0.0625 = 6.25\times 10^{-2}161​=0.0625=6.25×10−2

This is closest to

6×10−26\times 10^{-2}6×10−2
  1. Option check
  • A: 6×10−26\times 10^{-2}6×10−2 ✅
  • B: 3×10−33\times 10^{-3}3×10−3 ❌
  • C: 10−110^{-1}10−1 ❌
  • D: 6×1026\times 10^26×102 ❌

Therefore, the correct option is A.

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