#physics numerical

Superposition Principle and Equilibrium of Charges

Two point charges \( 5 \times 10^{-7} \, \text{C} \) and \( -7 \times 10^{-7} \, \text{C} \) are 150 cm apart in vacuum.

**Superposition principle** asserts net Coulomb force on charge equals vector sum of forces from each other charge independently, F_net = Σ F_i, where F_i = k q q_i/r_i² r̂_i. In equilateral triangle or square symmetry, components may cancel at centroid, producing equilibrium. Using Coulomb’s law: F = k (|q₁ q₂|/r²) . k = 9 × 10⁹ N·m²/C² , q₁ = 5 × 10⁻⁷ C , q₂ = -7 × 10⁻⁷ C , r = 1.5 m . |q₁ q₂| = 5 × 7 × 10⁻¹⁴ = 35 × 10⁻¹⁴ C² . r² = (1.5)² = 2.25 m² . F = 9 × 10⁹ × (35

Ref: NCERT > Physics Book > Electric Charges and Fields > Superposition Principle and Equilibrium of Charges

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Kepler's Laws and Planetary Motion

What is the gravitational potential due to Earth at 1.92×107m from its center? (ME\=6×1024kg,G\=6.67×10−11N m2/kg2)

U = −GMEr. U = −6.67×10−11×6×10241.92×107. U = −2.083×107J/kg. As per NCERT, applying relevant law/formula with correct units and sign convention leads to -2.1 × 10⁷ J/kg. This satisfies dimensional consistency and physical conditions given, so option B is scientifically correct.

Ref: Halliday & Resnick, Fundamentals of Physics, 11th ed., Chapter 13: Gravitation.

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Potential Energy of Mass Systems

What is the gravitational force on a 5kg mass 4m from the center of a spherical shell of mass 200kg and radius 3m? (G\=6

Outside shell: F = GMmr2. F = 6.67×10−11×200×542. F = 6.67×10−916≈4.17×10−10N. As per NCERT, applying relevant law/formula with correct units and sign convention leads to 4.2 × 10⁻¹⁰ N. This satisfies dimensional consistency and physical conditions given, so option C is scientifically correct.

Ref: Halliday & Resnick, Fundamentals of Physics, 11th ed., Chapter 13: Gravitation.

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Potential Energy of Mass Systems

What is the gravitational potential due to Earth at 4.48×107m from its center? (ME\=6×1024kg,G\=6.67×10−11N m2/kg2)

U = −GMEr. U = −6.67×10−11×6×10244.48×107. U = −8.936×106J/kg. As per NCERT, applying relevant law/formula with correct units and sign convention leads to -8.9 × 10⁶ J/kg. This satisfies dimensional consistency and physical conditions given, so option A is scientifically correct.

Ref: Halliday & Resnick, Fundamentals of Physics, 11th ed., Chapter 13: Gravitation.

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Potential Energy of Mass Systems

Three masses of 6kg each form an equilateral triangle with side 5m. What is the net force on one mass? (G\=6.67×10−11N m

Force between two masses: F = Gm2r2 = 6.67×10−116×652 = 9.61×10−11N. Two forces at 60°: FR = F2+F2+2F2cos⁡60∘. FR = (9.61×10−11)2(1+1+1) = 9.61×10−113. FR≈1.66×10−10N. As per NCERT, applying relevant law/formula with correct units and sign convention leads to 1.7 × 10⁻¹⁰ N. This satisfies dimensional consistency and physical conditions given, so option C is scientifically correct.

Ref: NCERT Class 11 Physics, Thermal Properties and Gravitation.

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Potential Energy of Mass Systems

How much energy is required to move a 200kg satellite from 6RE to 12RE from Earth’s center? (ME\=6×1024kg,RE\=6.4×106m,G

ΔE = −GMEm(1r2−1r1). r1 = 3.84×107m, r2 = 7.68×107m. ΔE = −6.67×10−11×6×1024×200(17.68×107−13.84×107). ΔE = −8.004×1016(−1.302×10−8)≈1.04×109J. As per NCERT, applying relevant law/formula with correct units and sign convention leads to 1.0 × 10⁹ J. This satisfies dimensional consistency and physical conditions given, so option A is scientifically correct.

Ref: Halliday & Resnick, Fundamentals of Physics, 11th ed., Chapter 13: Gravitation.

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Potential Energy of Mass Systems

Three 5kg masses form an equilateral triangle of side 4m. What is the potential energy of the system? (G\=6.67×10−11N m2

3 pairs: V = −3Gm2r. V = −36.67×10−11×5×54. V = −3×4.169×10−11 = −1.25×10−10J. As per NCERT, applying relevant law/formula with correct units and sign convention leads to -1.2 × 10⁻¹⁰ J. This satisfies dimensional consistency and physical conditions given, so option A is scientifically correct.

Ref: NCERT Class 11 Physics, Thermal Properties and Gravitation.

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Gravitational Potential and Field

Minimum speed to escape from 2RE from Earth’s center is? (g\=9.8m/s2,RE\=6.4×106m)

ve = 2gRE22RE = gRE. ve = 9.8×6.4×106 = 6.272×107. ve≈7.92×103m/s≈7.9km/s. As per NCERT, applying relevant law/formula with correct units and sign convention leads to 7.9 km/s. This satisfies dimensional consistency and physical conditions given, so option C is scientifically correct.

Ref: Halliday & Resnick, Fundamentals of Physics, 11th ed., Chapter 13: Gravitation.

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Gravitational Potential and Field

Three masses of 10kg each form an equilateral triangle with side 7m. What is the net force on one mass? (G\=6.67×10−11N

Force between two masses: F = Gm2r2 = 6.67×10−1110×1072 = 1.36×10−10N. Two forces at 60°: FR = F2+F2+2F2cos⁡60∘. FR = (1.36×10−10)2(1+1+1) = 1.36×10−103. FR≈2.36×10−10N. As per NCERT, applying relevant law/formula with correct units and sign convention leads to 2.4 × 10⁻¹⁰ N. This satisfies dimensional consistency and physical conditions given, so option C is scientifically correct.

Ref: NCERT Class 11 Physics, Thermal Properties and Gravitation.

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Gravitational Potential and Field

Three 6kg masses form an equilateral triangle of side 8m. What is the potential energy of the system? (G\=6.67×10−11N m2

3 pairs: V = −3Gm2r. V = −36.67×10−11×6×68. V = −3×3.0015×10−11 = −9.0×10−11J. As per NCERT, applying relevant law/formula with correct units and sign convention leads to -9.0 × 10⁻¹¹ J. This satisfies dimensional consistency and physical conditions given, so option B is scientifically correct.

Ref: NCERT Class 11 Physics, Thermal Properties and Gravitation.

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Gravitational Potential and Field

A projectile is launched at 3km/s from Earth’s surface. What is its maximum distance from the center? (Escape speed = 11

12vi2−ve22 = −ve22REr. 4.5−62.72 = −62.72REr. rRE = 62.7258.22≈1.077. r = 1.077×6.4×106≈6.89×106m. As per NCERT, applying relevant law/formula with correct units and sign convention leads to 6.9 × 10⁶ m. This satisfies dimensional consistency and physical conditions given, so option B is scientifically correct.

Ref: Halliday & Resnick, Fundamentals of Physics, 11th ed., Chapter 13: Gravitation.

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Universal Law of Gravitation

A satellite orbits a planet at 5×107m from its center with a period of 12 hours. What is the planet’s mass? (G\=6.67×10−

M = 4π2r3GT2. T = 12×3600 = 43200s, T2 = 1.866×109s2. r3 = (5×107)3 = 1.25×1023m3. M = 4×(3.14)2×1.25×10236.67×10−11×1.866×109. M = 4.93×10241.245×10−1≈3.96×1025kg. As per NCERT, applying relevant law/formula with correct units and sign convention leads to 4.0 × 10²⁵ kg. This satisfies dimensional consistency and physical conditions given, so option C is scientifically correct.

Ref: Halliday & Resnick, Fundamentals of Physics, 11th ed., Chapter 13: Gravitation.

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