Gravitational Potential Energy — Revision Notes

⚡ 30-Second Revision

GPE (near surface): — $U = mgh$ (reference $h=0$ at surface)
GPE (general): — $U = -\frac{GMm}{r}$ (reference $U=0$ at $r=infty$)
Gravitational Potential: — $V = U/m = -\frac{GM}{r}$
Work Done by External Agent: — $W_{ext} = Delta U = U_f - U_i$
Work Done by Gravity: — $W_g = -Delta U = U_i - U_f$
Conservation of Mechanical Energy: — $K_i + U_i = K_f + U_f$ (if no non-conservative forces)
Escape Velocity: — $v_{esc} = sqrt{\frac{2GM}{R}}$ (total energy at infinity is zero)
Total Energy in Circular Orbit: — $E = -\frac{GMm}{2r}$ (where $K = \frac{GMm}{2r}$ and $U = -\frac{GMm}{r}$)

2-Minute Revision

Gravitational Potential Energy (GPE) is the energy stored in a system of masses due to their relative positions in a gravitational field. For objects near Earth's surface, use the approximation $U = mgh$, where $h$ is height above a chosen zero level.

For larger distances, the general formula $U = -\frac{GMm}{r}$ is used, with zero potential energy defined at infinite separation. The negative sign signifies an attractive force and a bound system. Gravitational potential ($V$) is GPE per unit mass, $V = -\frac{GM}{r}$.

Work done by an external force to change an object's position against gravity equals the change in its GPE ($Delta U$). Work done by gravity is the negative of $Delta U$. A crucial concept is the conservation of mechanical energy ($K+U = \text{constant}$) in the absence of non-conservative forces.

This principle is vital for solving problems involving changes in speed and height, including those related to escape velocity and orbital motion. Remember that for a satellite in a circular orbit, its kinetic energy is half the magnitude of its potential energy, and its total energy is half its potential energy, with a negative sign.

5-Minute Revision

Gravitational Potential Energy (GPE) is a scalar quantity representing the energy stored in a system of masses due to their configuration in a gravitational field. It's fundamentally linked to the concept of work done by or against the conservative gravitational force.

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Near Earth's Surface: — For small heights $h$ compared to Earth's radius, GPE is approximated as $U = mgh$. Here, $g$ is considered constant, and the reference point ($h=0$) is typically the ground. This is a simplified model.

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General Formula: — For large distances or celestial mechanics, the GPE of two masses $M$ and $m$ separated by distance $r$ is $U = -\frac{GMm}{r}$. The reference point for zero potential energy is taken at infinity ($r \rightarrow infty$). The negative sign indicates that gravity is attractive and energy must be supplied to separate the masses. A more negative value means a more strongly bound system.

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Gravitational Potential: — This is GPE per unit mass, $V = U/m = -\frac{GM}{r}$. It describes the 'energy landscape' of the gravitational field itself.

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Work and GPE: — The work done by an external agent to move a mass $m$ from point A to point B against gravity (without changing kinetic energy) is $W_{ext} = U_B - U_A = Delta U$. The work done by the gravitational force itself is $W_g = U_A - U_B = -Delta U$.

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Conservation of Mechanical Energy: — In the absence of non-conservative forces (like air resistance), the total mechanical energy ($E = K + U$) of a system remains constant. This means $K_i + U_i = K_f + U_f$. This is a powerful tool for solving problems where objects move under gravity.

* Example: A ball of mass $m$ is dropped from height $h$ ($h ll R_E$). Initial $K_i=0$, $U_i=mgh$. Final $K_f=\frac{1}{2}mv^2$, $U_f=0$. So, $mgh = \frac{1}{2}mv^2 Rightarrow v = sqrt{2gh}$.

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Escape Velocity: — The minimum velocity required for an object to escape a planet's gravitational pull. It's derived by setting the total mechanical energy at infinity to zero: $rac{1}{2}mv_{esc}^2 + (-\frac{GMm}{R}) = 0 Rightarrow v_{esc} = sqrt{\frac{2GM}{R}}$.

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Orbital Energy: — For a satellite in a circular orbit of radius $r$, its kinetic energy is $K = \frac{GMm}{2r}$, and its potential energy is $U = -\frac{GMm}{r}$. The total mechanical energy is $E = K+U = -\frac{GMm}{2r}$. This negative total energy confirms it's a bound orbit.

Key takeaway: Always consider the reference point for GPE and whether the $mgh$ approximation is valid. The conservation of mechanical energy is your primary problem-solving tool.

Prelims Revision Notes

Gravitational Potential Energy (GPE) is a scalar quantity representing the energy stored in a system of masses due to their relative positions. It is a fundamental concept in gravitation.

1. Definition and Formulas:

Near Earth's Surface: — For objects at height $h$ above the surface (where $h ll R_E$), $U = mgh$. Here, $g$ is assumed constant, and $U=0$ at $h=0$ (surface).
General Formula: — For any two point masses $M$ and $m$ separated by a distance $r$, $U = -\frac{GMm}{r}$. Here, $U=0$ is defined at $r=infty$. The negative sign indicates an attractive force and a bound system.

2. Gravitational Potential ($V$):

Potential energy per unit mass: $V = U/m$.
For a point mass $M$ at distance $r$: $V = -\frac{GM}{r}$.
Units: J/kg.

3. Work Done:

Work done by an external agent to move mass $m$ from $r_1$ to $r_2$: $W_{ext} = U_2 - U_1 = Delta U$.
Work done by gravitational force: $W_g = U_1 - U_2 = -Delta U$.

4. Conservation of Mechanical Energy:

In the absence of non-conservative forces (like air resistance), total mechanical energy ($E = K + U$) is conserved: $K_i + U_i = K_f + U_f$.
This is crucial for problems involving changes in height and speed.

5. Escape Velocity ($v_{esc}$):

Minimum velocity required to escape a gravitational field. Total energy at infinity is zero.
$rac{1}{2}mv_{esc}^2 + (-\frac{GMm}{R}) = 0 Rightarrow v_{esc} = sqrt{\frac{2GM}{R}}$.
On Earth's surface, $v_{esc} = sqrt{2gR_E} approx 11.2,\text{km/s}$.

6. Orbital Energy of a Satellite:

For a circular orbit of radius $r$ around mass $M$:

* Kinetic Energy: $K = \frac{GMm}{2r}$ * Potential Energy: $U = -\frac{GMm}{r}$ * Total Energy: $E = K + U = -\frac{GMm}{2r}$

Note: $K = -E$ and $U = 2E$.

7. Potential Energy of a System of Particles:

Sum of potential energies of all unique pairs of particles.
For three particles $m_1, m_2, m_3$: $U_{total} = -\frac{Gm_1m_2}{r_{12}} - \frac{Gm_1m_3}{r_{13}} - \frac{Gm_2m_3}{r_{23}}$.

Common Traps:

Forgetting the negative sign in $U = -\frac{GMm}{r}$.
Using $mgh$ when $h$ is comparable to $R_E$.
Confusing gravitational potential with potential energy.
Sign errors in work done calculations.

Vyyuha Quick Recall

Gravity Pulls, Energy Negative, Infinity Zero.

Gravity Pulls: Reminds you gravity is attractive.
Energy Negative: Helps recall the negative sign in the general formula $U = -\frac{GMm}{r}$.
Infinity Zero: Reminds you that the reference point for zero potential energy is at infinity.