Gravitational PE — Revision Notes

⚡ 30-Second Revision

GPE near Earth: — $U = mgh$ (reference $h=0$ at ground).
Universal GPE: — $U = -\frac{GMm}{r}$ (reference $r=\infty$ where $U=0$).
Change in GPE: — $\Delta U = U_f - U_i = mg(h_f - h_i)$ (near Earth).
Work done by gravity: — $W_g = -\Delta U$.
Conservative Force: — Gravity is conservative; work done is path-independent.
Gravitational Potential: — $V = U/m = -\frac{GM}{r}$ (J/kg).
Total Mechanical Energy: — $E = K + U = \text{constant}$ (if only conservative forces act).
Escape Velocity: — $v_e = \sqrt{\frac{2GM}{R}}$ (from surface of planet R).

2-Minute Revision

Gravitational Potential Energy (GPE) is the energy an object possesses due to its position in a gravitational field. It's a scalar quantity and is always defined relative to a chosen zero reference point.

For objects near Earth's surface, GPE is approximated as $U = mgh$, where 'h' is height above a reference (often the ground). For universal gravitation, between two masses M and m separated by distance 'r', GPE is $U = -\frac{GMm}{r}$, with infinity as the zero reference.

The negative sign indicates an attractive force and a bound system. The *change* in GPE, $\Delta U$, is crucial for problem-solving and is independent of the reference point. Gravity is a conservative force, meaning the work done by it is path-independent, allowing for the definition of GPE.

This also implies that in the absence of non-conservative forces, total mechanical energy ($K+U$) is conserved. Remember to distinguish GPE from gravitational potential, which is GPE per unit mass.

5-Minute Revision

Gravitational Potential Energy (GPE) is a form of stored energy resulting from the position of an object within a gravitational field. It's fundamentally linked to the concept of conservative forces, where the work done by the force is independent of the path taken. For gravity, this means we can assign a unique potential energy value to each position.

There are two primary formulas for GPE:

1
Near Earth's Surface: — $U = mgh$. This is valid when the height 'h' is small compared to the Earth's radius, allowing us to assume constant gravitational acceleration 'g'. The reference point for $h=0$ is arbitrary, often chosen as the ground or the lowest point in a problem. If an object is below this reference, 'h' becomes negative, and so does GPE.
2
Universal Gravitational Potential Energy: — $U = -\frac{GMm}{r}$. This general formula applies to any two masses M and m separated by a distance 'r' (measured from their centers). Here, the standard reference point for zero potential energy is infinity ($r=\infty$), where the gravitational force is negligible. The negative sign signifies that gravity is an attractive force and that the system is bound; energy must be supplied to separate the masses. A more negative GPE implies a more tightly bound system.

Key Concepts for NEET:

Conservation of Mechanical Energy: — In the absence of non-conservative forces (like friction), the total mechanical energy ($K+U$) of a system remains constant: $K_i + U_i = K_f + U_f$. This is a powerful tool for solving problems involving falling objects, projectiles, and orbital motion.
Work-Energy Relation: — The work done by gravity is $W_g = -\Delta U$. If GPE decreases, gravity does positive work.
Gravitational Potential (V): — This is GPE per unit mass, $V = U/m = -\frac{GM}{r}$. It's a property of the gravitational field itself, measured in J/kg. Don't confuse it with GPE.
Escape Velocity: — The minimum initial velocity required for an object to completely escape a planet's gravitational pull, meaning its total mechanical energy becomes zero at infinity. $v_e = \sqrt{\frac{2GM}{R}}$.

Worked Example: A $1,\text{kg}$ ball is thrown upwards with an initial speed of $10,\text{m/s}$ from the ground. What is its GPE at its maximum height? (Take $g = 10,\text{m/s}^2$)

Solution: — By conservation of energy, initial kinetic energy converts entirely to GPE at maximum height.

* Initial Kinetic Energy ($K_i$) = $\frac{1}{2}mv^2 = \frac{1}{2} \times 1,\text{kg} \times (10,\text{m/s})^2 = 50,\text{J}$. * At maximum height, $K_f = 0$. * So, GPE at maximum height ($U_f$) = $K_i = 50,\text{J}$. * Alternatively, find max height $H = v^2/(2g) = 10^2/(2 \times 10) = 5,\text{m}$. Then $U_f = mgH = 1 \times 10 \times 5 = 50,\text{J}$.

Prelims Revision Notes

1
Definition: — Gravitational Potential Energy (GPE) is energy stored due to position in a gravitational field. It's a scalar quantity.
2
Reference Point: — GPE is always relative to a chosen zero potential energy level. The *change* in GPE is independent of this choice.
3
GPE near Earth's surface: — $U = mgh$.

* 'm' = mass (kg), 'g' = acceleration due to gravity ($9.8,\text{m/s}^2$ or $10,\text{m/s}^2$), 'h' = height above reference (m). * Reference ($h=0$) is typically the ground or lowest point. * Can be positive (above reference) or negative (below reference).

1
Universal GPE: — $U = -\frac{GMm}{r}$.

* 'G' = Universal Gravitational Constant ($6.67 \times 10^{-11},\text{Nm}^2/\text{kg}^2$). * 'M', 'm' = masses (kg), 'r' = distance between centers of masses (m). * Reference ($r=\infty$) is where $U=0$. * Always negative, indicating an attractive force and a bound system. More negative means stronger binding.

1
Gravitational Potential (V): — $V = U/m = -\frac{GM}{r}$. It's GPE per unit mass, measured in J/kg.
2
Work Done by Gravity: — $W_g = -\Delta U$. If GPE decreases, gravity does positive work.
3
Conservation of Mechanical Energy: — If only conservative forces (like gravity) act, total mechanical energy ($E = K + U$) is conserved: $K_i + U_i = K_f + U_f$.
4
Escape Velocity: — Minimum speed to escape a gravitational field. $v_e = \sqrt{\frac{2GM}{R}}$. Derived by setting $K_i + U_i = 0$ (total energy at infinity).
5
Common Mistakes:

* Confusing 'h' with 'r' in universal GPE formula ($r = R_E + h$). * Ignoring the negative sign in universal GPE calculations. * Mixing up GPE and gravitational potential. * Incorrectly applying conservation of energy when non-conservative forces are present.

Vyyuha Quick Recall

GPE: Gravity's Position Energy. Remember Many Good Heights ($mgh$) for near Earth, and Giant Masses Minus Radius ($-GMm/r$) for universal. The negative sign means it's Nice and Negative when Near.