Question

For a system of \(N\) particles subject to a uniform gravitational field g acting vertically down, prove that the total gravitational potential energy is the same as if all the mass were concentrated at the center of mass of the system; that is, \(U=\sum_{\alpha} U_{\alpha}=M g Y\) where \(M=\sum m_{\alpha}\) is the total mass and \(\mathbf{R}=(X, Y, Z)\) is the position of the \(\mathrm{CM},\) with the \(y\) coordinate measured vertically up. [Hint: We know from Problem 4.5 that \(\left.U_{\alpha}=m_{\alpha} g y_{\alpha} .\right]\)

Short answer

The total gravitational potential energy is \( U = M g Y \), as if all mass were at the center of mass.

Step-by-step solution

Define Potential Energy

The potential energy of a single particle in a gravitational field is given by \( U_{\alpha} = m_{\alpha} g y_{\alpha} \), where \( m_{\alpha} \) is the mass of the particle and \( y_{\alpha} \) is its vertical position.

Express Total Potential Energy

The total gravitational potential energy for the system is the sum of the potential energies of all particles: \( U = \sum_{\alpha} U_{\alpha} = \sum_{\alpha} m_{\alpha} g y_{\alpha} \).

Define Center of Mass

The vertical coordinate of the center of mass is defined as \( Y = \frac{\sum m_{\alpha} y_{\alpha}}{M} \), where \( M = \sum m_{\alpha} \) is the total mass of the system.

Substitute Center of Mass

Multiply both sides of the expression for \( Y \) by \( M \) to get: \( M Y = \sum m_{\alpha} y_{\alpha} \).

Relate Total Potential Energy to Center of Mass

Substitute \( \sum m_{\alpha} y_{\alpha} = M Y \) into the expression for total potential energy: \( U = g \sum m_{\alpha} y_{\alpha} = g M Y \).

Conclude

This expression \( U = M g Y \) shows that the total potential energy is the same as if all the mass were concentrated at the center of mass at height \( Y \).

Key concepts

Center of Mass

The center of mass is a crucial concept in physics and plays a pivotal role in understanding gravitational systems. Essentially, the center of mass is a point where the total mass of a system appears to be concentrated. For any system of particles, we use the center of mass to simplify complex calculations and make predictions about motion.
The vertical coordinate of the center of mass, denoted as \( Y \), is calculated using the formula: - \( Y = \frac{\sum m_{\alpha} y_{\alpha}}{M} \).
This means that the center of mass accounts for all of the individual masses \( m_\alpha \) and their respective positions \( y_\alpha \) in the system. The sum of these multiplied terms is then divided by the total mass \( M \).
In practical terms, when observing an object like a seesaw, the point at which it balances perfectly on its pivot is where its center of mass is located. This principle is applied to any system of particles to simplify the calculation of its dynamics under gravitational forces.

Uniform Gravitational Field

In physics, a uniform gravitational field is a theoretical region of space where a gravitational field has the same force acting everywhere. This means that any object within this field experiences the same gravitational pull regardless of its position.
The gravitational field strength is denoted by \( g \), commonly recognized as the acceleration due to gravity on Earth's surface, approximately \( 9.8 \, \text{m/s}^2 \). This constant force is crucial for calculating gravitational potential energy.
When dealing with systems of particles, assuming a uniform gravitational field simplifies the problem significantly. It allows us to calculate potential energy using the formula: - \( U_{\alpha} = m_\alpha g y_\alpha \). This means each particle's potential energy depends solely on its mass \( m_\alpha \), the gravitational field strength \( g \), and its vertical position \( y_\alpha \).
Understanding uniform gravitational fields helps us model real-world phenomena and develop insights into how objects interact with gravitational forces in a straightforward and effective manner.

Total Mass

Total mass is a fundamental concept that refers to the sum of all individual masses in a given system. It serves as a single value that represents the entire mass content of the system. In the context of gravitational potential energy, the total mass plays a significant role.
The total mass \( M \) is calculated using:- \( M = \sum m_{\alpha} \), where each \( m_\alpha \) represents the mass of individual particles within the system.
This collective mass is used to simplify calculations by allowing us to treat the entire system as if all mass were concentrated at a single point, specifically the center of mass. This simplifies the expression for gravitational potential energy to \( U = MgY \), where \( Y \) is the vertical position of the center of mass.
Recognizing the importance of total mass helps us understand complex interactions in multi-particle systems. It lets us analyze system behavior as a whole, rather than focusing on individual particles separately, making it easier to predict and explain physical phenomena efficiently.