Question

If Wa, Wb, and Wc represent the work done in moving a particle from \(\mathrm{X}\) to \(\mathrm{Y}\) along three different path \(\mathrm{a}, \mathrm{b}\), and \(\mathrm{c}\) respectively (as shown) in the gravitational field of a point mass \(\mathrm{m}\), find the correct relation between \(\mathrm{Wa}, \mathrm{Wb}\) and \(\mathrm{Wc}\) (A) \(\mathrm{Wb}>\mathrm{Wa}>\mathrm{Wc}\) (B) \(\mathrm{Wa}<\mathrm{Wb}<\mathrm{Wc}\) (C) \(\mathrm{Wa}>\mathrm{Wb}>\mathrm{Wc}\) (D) \(\mathrm{Wa}=\mathrm{Wb}=\mathrm{Wc}\)

Short answer

The correct relation between Wa, Wb, and Wc is \(W_a = W_b = W_c\), as gravitational force is a conservative force and does not depend on the path taken.

Step-by-step solution

Define the Work Done Formula for Gravitational Force

When dealing with gravitational forces, the force acting on an object is given by: \[F = \frac{Gm1m2}{r^2}\] where F is the force acting on the object, G is the gravitational constant, m1 and m2 are the masses being considered, and r is the distance between the centers of mass. Due to the nature of gravitational forces, they are conservative forces and the work done does not depend on the path taken. To compute the work done in moving a particle from point X to Y in the gravitational field, we will use the following formula: \(W = -\int_{x_i}^{x_f} F \cdot dr\) where W is the work done, xi and xf are the initial and final positions along the particle's path, F is the gravitational force acting on the object, and dr is the differential displacement.

Calculate work done for all paths

Since gravitational force is a conservative force, the work done moving an object from point X to point Y will not depend on the path taken. This means that regardless of whether we choose path a, b, or c, the work done will always be the same. Let's calculate the work done for each path. For path a: \(W_a = -\int_{x_i}^{x_f} F \cdot dr\) For path b: \(W_b = -\int_{x_i}^{x_f} F \cdot dr\) For path c: \(W_c = -\int_{x_i}^{x_f} F \cdot dr\) Notice that all three work-done equations are the same, which is expected for a conservative force like the gravitational force.

Determine the relationship between Wa, Wb, and Wc

Now that we've calculated the work done for each path, we can easily determine the relationship between them. Since we obtained the same equation for the work done in all three paths, it follows that: \(W_a = W_b = W_c\) This means, the correct relation between Wa, Wb, and Wc is given by option (D).

Key concepts

Gravitational Force

Gravitational force is the attractive force exerted by two masses on each other. It is one of the universal forces that acts over large distances without requiring contact between the objects. The formula for calculating gravitational force is given by:

\[ F = \frac{Gm_1m_2}{r^2} \]
where:

• \(F\) is the gravitational force,
• \(G\) is the gravitational constant (approximately \(6.674 \times 10^{-11} \text{N m}^2/\text{kg}^2\)),
• \(m_1\) and \(m_2\) are the masses of the two objects,
• \(r\) is the distance between the centers of the two masses.

This force is always attractive and directs towards the other mass along the line joining the centers of the two masses. Despite its small size on an everyday scale, gravitational force plays a crucial role in celestial phenomena and keeps planets in orbit around stars.

Work Done in Gravitational Fields

Work done in gravitational fields refers to the energy required to move a mass in the presence of a gravitational force. Importantly, gravitational force is a type of conservative force. This means that the work done by or against it is independent of the path taken and depends only on the initial and final positions in the gravitational field.

To calculate the work done in a gravitational field:\[ W = -\int_{x_i}^{x_f} F \cdot dr \]
In this formula:

• \(W\) represents the work done,
• \(F\) is the gravitational force,
• \(dr\) is the differential displacement along the path,
• \(x_i\) and \(x_f\) are the initial and final positions, respectively.

The negative sign indicates that work is done by the gravitational field when an object is moved to a higher gravitational potential energy state.

Path Independence

A significant attribute of conservative forces, such as gravitational force, is path independence. This means that the work done by a gravitational field when moving an object from one point to another is the same regardless of the path taken.

This concept allows us to make calculations much simpler, as only the initial and final positions matter. For instance, if you were to move a particle in the gravitational field of a point mass along three different paths

• Path a,
• Path b,
• Path c,

all would require the same amount of work (given that the starting and ending points are the same).

This property is a hallmark of all conservative forces, making it easier to analyze energy transformations in many physical systems.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. It is an essential concept when discussing conservative forces and path independence.

The formula for gravitational potential energy is:\[ U = -\frac{Gm_1m_2}{r} \]
where:

• \(U\) is the gravitational potential energy,
• \(G\) is the gravitational constant,
• \(m_1\) and \(m_2\) are the respective masses of the objects,
• \(r\) is the distance between their centers.

Here, the negative sign indicates that the energy is lower when the objects are closer together. As objects are moved closer in a gravitational field, they need to gain energy to "escape" or separate, aligning with the concept that work is done by the gravitational field.

Gravitational potential energy is foundational in understanding many astronomical phenomena and is pivotal in calculating the work done in moving objects through gravitational fields.