Q.Evaluate: the definite integral of (GMm / x^2) dx, taken from x = R to x = infinity (i.e. Integral[R to infinity] (GMm/x^2) dx).
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Gravitational Potential Energy
The Intuition: Energy Stored by Height
Imagine holding a heavy book above the floor. Your arm feels tired — that's because you're working against gravity. If you let go, the book falls and gains speed. Where did that motion come from? It came from the position of the book. By lifting it, you stored energy in the Earth–book system. That stored energy is gravitational potential energy.
The higher you lift, the more energy you store. The heavier the object, the more energy you store. This is the core idea: Gravitational potential energy is the energy an object has because of its position in a gravitational field.
The Precise Definition
Gravitational potential energy (U) is the work done against gravity to bring an object from a reference point (usually the ground) to its current position.
For objects near the Earth's surface (where gravity is roughly constant), the formula is beautifully simple:
U=mgh
Where:
- U = gravitational potential energy (joules, J)
- m = mass of the object (kg)
- g = acceleration due to gravity (≈ 9.8 m/s² on Earth)
- h = height above the reference point (m)
Why "Potential"?
The word "potential" means "stored and ready to be used." The book at height h has the potential to do work — it can smash a table, compress a spring, or generate sound when it hits the ground. That energy was put in when you lifted it.
The Reference Point is Arbitrary
Here's a crucial point: Only changes in gravitational potential energy matter. You can choose any height as h=0. In most problems, we take the ground as zero, but you could take the floor, the tabletop, or even the ceiling.
If you lift a 2 kg book from the floor (h=0) to a shelf (h=2 m), the change in potential energy is:
ΔU=mgΔh=2×9.8×2=39.2 J
If you instead took the shelf as h=0, the book on the floor would have negative potential energy (−39.2 J). The difference between the two positions is still 39.2 J — that's what matters.
Never say "the object has mgh energy" without specifying the reference level. The value is meaningless without a zero point.
The Bigger Picture: Variable Gravity
The formula U=mgh works only when g is constant — that is, near Earth's surface. For large distances (like a rocket leaving Earth), gravity weakens with distance. The general formula for gravitational potential energy between two masses M and m separated by distance r is:
U=−rGMm
The negative sign means that potential energy is zero at infinite separation and becomes more negative as objects come closer. This is the true definition, and U=mgh is a special case of it (derived by approximating near the surface).
Key Takeaways for Exams
- Gravitational potential energy is always relative — you must state or imply a reference level. …
This is the standard integral for the work needed to remove a mass from radius R to infinity against gravity. …
Integrating GMm/x^2 with respect to x from R to infinity gives GMm/R -- this is the magnitude of work required to take a mass m from the surface (radius R) of a body of mass M to infinity against gravity.
We evaluate:
Integral[R to infinity] (GMm/x^2) dx
Since GMm is a constant, take it outside:
= GMm * Integral[R to infinity] x^-2 dx
= GMm * [ -1/x ] from R to infinity
= GMm * [ (-1/infinity) - (-1/R) ]
= GMm * [ 0 + 1/R ]
= GMm/R
…
Showing the 12 most recent of 16 on this concept.
- CBSE 2026Set ANNUAL1 markMCQQ.Relation between orbital velocity (Vo) of a satellite revolving near the earth and escape velocity (Ve) of a body from the earth's surface is(a) Ve = √2 Vo(b) Vo = √2 Ve(c) Ve = √(Vo/2)(d) Vo = √(Ve/2)
›Reveal solutionSolution
Ve = sqrt(2) Vo. Answer (A).
Orbital velocity of a satellite close to the earth: Vo = sqrt(gR).
Escape velocity from the earth's surface: Ve = sqrt(2gR).
…
- CBSE 2025Set ANNUAL1 markMCQQ.Gravitational potential at a distance r from a body of mass M is(a) GM / r(b) GM / r^2(c) -GM / r(d) -GM / r^2
›Reveal solutionSolution
Gravitational potential V(r) = -GM/r; the negative sign reflects that gravity does positive work as a mass falls inward from infinity (where potential is taken as zero).
Gravitational potential at a point is defined as the work done per unit mass in bringing a test mass from infinity to that point, against/with the gravitational field:
V(r) = -Integral (from infinity to r) of (GM/r^2) dr = -GM/r
…
- CBSE 2025Set ANNUAL1 markQ.Answer in one word or one sentence: Calculate the gravitational potential energy of a stone of mass 5 kg placed at a height 10 m above the earth surface. (g = 9.8 m/s^2)
›Reveal solutionSolution
Using PE = mgh, a 5 kg stone at 10 m height has gravitational potential energy of 490 J.
Near the Earth's surface, gravitational potential energy (relative to the surface) is given by PE = mgh, where m is mass, g is acceleration due to gravity, and h is height above the reference level. …
- CBSE 2024Set ANNUAL1 markMCQQ.The gravitational potential inside a hollow sphere (A) is not uniform (B) is uniform (C) is zero (D) none of these
›Reveal solutionSolution
Gravitational potential inside a hollow sphere is uniform (constant), equal to its surface value.
For a thin uniform spherical shell, the gravitational field at any point inside the cavity is exactly zero (contributions from all parts of the shell cancel). Since E=−dV/dr, a zero field means V does not vary with position inside the shell — the potent …
- CBSE 2024Set ANNUAL1 markMCQQ.The gravitational potential V at a point P, outside a solid sphere is V = −GM/x, where G is gravitational constant and M is mass of the sphere. 'x' represents (A) radius of the sphere (B) distance of point P from the surface of the sphere (C) distance of point P from the centre of the sphere (D) square of distance of point P from the centre of the sphere
›Reveal solutionSolution
In V=−GM/x, x is measured from the centre of the sphere.
For a uniform solid sphere, treated as a point mass for external points (by the shell theorem), the gravitational potential at a distance x from its centre is V=−GM/x, valid for x≥R (the sphere's radius). The variable x is always the centre-to-point distance, not the surface-to …
- CBSE 2023Set ANNUAL1 markQ.If the earth revolves around the Sun in a circular orbit, then what would be the work done by the gravitational force?
›Reveal solutionSolution
Work done by a force is W=F⋅d; since gravity on Earth always points radially toward the Sun while Earth's velocity (and hence its displacement) is always tangential to the circular path, the force is always perpendicular to the motion, so W=0.
For a body moving in a circle, work done by any force is dW=F⋅ds=Fdscosθ, where θ is the angle between the force and the displacement. The Sun's gravitational pull on the Earth acts along the radius, i.e. it is centripetal — always directed from the Earth toward the Sun. The Earth's instantaneous displacement at any point of a circular orbit is always tangent to the circle, i.e. perpendicular to the radius. So the angle between the gravitational force and the displacement is always θ=90∘, and cos90∘=0. Hence the elementary work done in each small displacement is zero, and integrating over the whole o …
- CBSE 2023Set ANNUAL1 markMCQQ.Gravitational potential inside a hollow spher is:(a) not uniform(b) uniform(c) zero(d) none of these
›Reveal solutionSolution
Inside a hollow sphere the gravitational potential is uniform.
By the shell theorem, the gravitational field inside a uniform hollow sphere is zero everywhere. Since the field E = −dV/dr = 0, the potential V does not change with position: it is con …
- CBSE 2023Set ANNUAL1 markMCQQ.Escape velocity for the object from the earth is:(a) 11.2 km/s(b) 11.2 m/s(c) 112 km/s(d) 112 km/h
›Reveal solutionSolution
Earth's escape velocity ≈ 11.2 km/s.
Escape velocity v_e = √(2gR) = √(2GM/R). Using g = 9.8 m/s^2 and R = 6.4 × 10^6 m, …
- CBSE 2023Set ANNUAL1 markMCQQ.If R is the radius of the earth and g is acceleration due to granity, then expression for escape velocity is:(a) R/g(b) 2R/g(c) √(2Rg)(d) 2Rg
›Reveal solutionSolution
Escape velocity = √(2gR).
To just escape Earth's gravity, the kinetic energy given must equal the gravitational potential energy magnitude: …
- CBSE 2022Set ANNUAL1 markMCQQ.Gravitational force is:(a) a conservative force(b) Pseudo force(c) a non-conservative force(d) None of the above
›Reveal solutionSolution
Gravitational force is conservative because the work it does depends only on the start and end points, not the path.
Concept. A force is called conservative if the work done by it in moving an object between two points is independent of the path taken, and depends only on the initial and final positions. Equivalently, the work done over any closed path is zero, and the force can be derived from a potential energy function (F=−drdU).
…
- CBSE 2022Set ANNUAL1 markMCQQ.A body enters in the gravitational field of the earth from external atmosphere. The potential energy of the earth-body system:(a) will increase(b) will decrease(c) will remain unchanged(d) None of the above
›Reveal solutionSolution
As a body falls toward Earth from far away, gravitational PE decreases (becomes more negative) while KE increases.
Reasoning. Gravitational potential energy between Earth (mass M) and a body (mass m) at separation r is:
U(r)=−rGMm
This is taken as zero at r=∞ and becomes increasingly negative as r decreases (i.e. as the body gets closer to Earth).
…
- CBSE 2022Set sz1 markQ.Evaluate: the definite integral of (GMm / x^2) dx, taken from x = R to x = infinity (i.e. Integral[R to infinity] (GMm/x^2) dx).
›Reveal solutionSolution
Integrating GMm/x^2 with respect to x from R to infinity gives GMm/R -- this is the magnitude of work required to take a mass m from the surface (radius R) of a body of mass M to infinity against gravity.
We evaluate:
Integral[R to infinity] (GMm/x^2) dx
Since GMm is a constant, take it outside:
= GMm * Integral[R to infinity] x^-2 dx
= GMm * [ -1/x ] from R to infinity
= GMm * [ (-1/infinity) - (-1/R) ]
= GMm * [ 0 + 1/R ]
= GMm/R
…
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