Question

Can the expression for gravitational potential energy \(U_{g}(y)=m g y\) be used to analyze high-altitude motion? Why or why not?

Short answer

Why or why not? Answer: No, the expression \(U_{g}(y) = mgy\) cannot be used to accurately analyze high-altitude motion, as it is based on the assumption of a uniform gravitational field which is only applicable for low-altitude motion. At high altitudes, the gravitational force decreases with altitude, and the accurate equation for gravitational potential energy to be used is \(U_g(y) = -G\frac{Mm}{R+y}\), where G is the gravitational constant, M is Earth's mass, and R is Earth's radius.

Step-by-step solution

Understanding Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. The energy is a conservative and scalar quantity that depends on the object's mass (m), the distance from the reference point (usually the Earth's surface) at height y, and the gravitational acceleration (g). The equation for gravitational potential energy is given by \(U_g(y) = mgy\).

Applicability of the Expression

The expression \(U_g(y) = mgy\) is an approximation that is valid for low-altitude motion, as it is derived under the assumption that the gravitational field is uniform. The reason for the uniformity of the gravitational field is that Earth's radius is considered much larger than the altitude at which the object is located.

Limitations of the Expression at High Altitude

When an object is at a considerable altitude (i.e., Earth's radius is no longer much larger than the altitude), the gravitational field becomes non-uniform, and the gravitational force decreases with altitude. The expression \(U_{g}(y) = mgy\) fails to take this effect into account, making it less accurate for analyzing high-altitude motion.

Considering the Gravitational Force at High Altitude

To analyze high-altitude motion, the gravitational force should be considered as the force acting between two masses, the Earth's mass (M) and the object's mass(m), separated by a distance R (Earth's radius) and the object's altitude y. The force between these masses is described using Newton's Law of Universal Gravitation: \(F_g = G\frac{Mm}{(R+y)^2}\), where G is the gravitational constant. Taking into account this force, the gravitational potential energy equation for high altitude becomes: \(U_g(y) = -G\frac{Mm}{R+y}\)

Conclusion

To sum up, the expression \(U_g(y) = mgy\) cannot be used to accurately analyze high-altitude motion, due to the assumptions used in its derivation. For high-altitude motion, we need to consider the full equation of gravitational potential energy with the variable gravitational force, which is \(U_g(y) = -G\frac{Mm}{R+y}\).

Key concepts

Uniform Gravitational Field

The idea of a uniform gravitational field simplifies the mathematical treatment of gravity near the Earth's surface. In this model, the field exerts the same amount of force per unit mass on all objects, regardless of their location within the field.

Physically, this assumes that the Earth's gravitational pull does not vary in strength over the distance of the object's motion. The essential formula, which students often encounter in physics problems, is given by the equation
\( U_g(y) = mgy \). Here, m represents the mass, g is the acceleration due to gravity (approximately 9.81 m/s² near the Earth's surface), and y is the height above a reference level.

However, this equation assumes that g remains constant with altitude, which is a reasonable approximation only for relatively small increases in height compared to the Earth's radius. For motions that occur at much higher altitudes, this assumption no longer holds, and the concept of a uniform gravitational field becomes inaccurate.

High-Altitude Motion

When discussing high-altitude motion, we must consider the changes in the Earth’s gravitational force as an object moves further away from the Earth's surface. As altitude increases, the force of gravity decreases, which is not accounted for in the uniform gravitational field model.

The variations in gravitational pull at higher altitudes mean that the neat linear equation for gravitational potential energy doesn't quite fit the actual conditions. Thus, the simplification \( U_g(y) = mgy \) should not be used when analyzing objects, like satellites, that travel at altitudes where the Earth's curvature and the decrease in gravitational force become significant. Instead, a more complex model that accounts for the distance between the object and the Earth's center is needed to accurately describe the potential energy of high-altitude objects.

Newton's Law of Universal Gravitation

Sir Isaac Newton's Law of Universal Gravitation provides the foundation for understanding gravitational interactions between objects. This powerful law states that every point mass attracts every other point mass by a force acting along the line intersecting both points.

The formula derived from this law, \( F_g = G\frac{Mm}{(R+y)^2} \) (where F_g is the gravitational force, G is the gravitational constant, M and m are the masses of the Earth and the object respectively, R is the Earth's radius, and y is the altitude above the Earth's surface), enables the calculation of the gravitational force at any distance. This law becomes particularly important for analyzing motion at high altitudes, where the variation of g with altitude must be accounted for.

Conservative Scalar Quantity

Gravitational potential energy is referred to as a conservative scalar quantity. It's 'conservative' because the total energy is conserved when an object moves in a gravitational field, disregarding other forces or energy losses. No matter the path taken between two points, the change in gravitational potential energy will remain the same.

Being a 'scalar' quantity means it does not have direction, unlike vector quantities which describe magnitude and direction. Gravitational potential energy only has magnitude, which is a measure of the work done against gravity to move an object to a certain position. This scalar nature simplifies the calculation of energy in gravitational fields, as only the magnitude of the displacement, not the direction, affects the potential energy of an object. Consequently, understanding this concept helps students realize why the potential energy can be calculated independently of the path taken by the object.