Problem 74 - University Physics with Modern Physics

Question

A uniform wire with mass \(M\) and length \(L\) is bent into a semicircle. Find the magnitude and direction of the gravitational force this wire exerts on a point with mass \(m\) placed at the center of curvature of the semicircle.

Step-by-Step Solution

Answer

The force magnitude is \( \frac{2GmM\pi}{L^2} \), directed vertically towards the wire.

Step 1: Understanding the Problem

We need to find the gravitational force exerted by a semicircular wire of mass \( M \) and length \( L \) on a point mass \( m \) located at the center of its curvature.

Step 2: Determine Linear Mass Density

Since the wire is uniform, its linear mass density \( \lambda \) can be given by \( \lambda = \frac{M}{L} \). This will allow us to express small masses along the wire in terms of arc length.

Step 3: Select a Small Element of the Wire

Consider a small element of the wire with arc length \( ds \) and mass \( dm = \lambda \, ds \). This element exerts a gravitational force \( dF \) on the mass \( m \).

Step 4: Express Differential Force

The small element creates a force \( dF = \frac{G \, m \, dm}{R^2} = \frac{G \, m \, \, \lambda \, ds}{R^2} \) toward the center, where \( R \) is the radius of the semicircle. Given \( L = \pi R \), we find \( R = \frac{L}{\pi} \).

Step 5: Resolve Differential Force Components

Due to symmetry, horizontal components of \( dF \) (along the x-axis) cancel out. Vertical components add up. Each component \( dF_y = dF \sin \theta = \frac{G \, m \, \lambda \, ds}{R^2} \sin \theta \), with \( \theta \) as the angle from the center.

Step 6: Integrate Over the Semicircle

Since symmetry cancels horizontal forces, integrate the vertical components: \( F_y = \int_0^\pi \frac{G \, m \, \lambda}{R^2} \sin \theta \, R \, d\theta \). Substitute \( \lambda = \frac{M}{\pi R} \) and \( R = \frac{L}{\pi} \).

Step 7: Solve the Integral

Substitute and evaluate the definite integral: \( F_y = \frac{G \, m \, M}{R^2 \pi} \int_0^\pi \sin \theta \, d\theta = \frac{G \, m \, M}{R^2 \pi} [1 - (-1)] = \frac{2 \, G \, m \, M}{R^2 \pi} \).

Step 8: Insert Known Radius

Substituting \( R = \frac{L}{\pi} \) in the force expression: \( F_y = \frac{2 \, G \, m \, M \pi^2}{L^2 \pi} = \frac{2 \, G \, m \, M \pi}{L^2} \). Hence, the force is directed vertically towards the wire.

Key Concepts

Understanding Linear Mass Density

Linear mass density is a fundamental concept when dealing with objects like wires and rods. For any uniform object, linear mass density, denoted as \( \lambda \), is defined as the mass per unit length. It's calculated by dividing the total mass \( M \) of the object by its total length \( L \), so \( \lambda = \frac{M}{L} \). This concept allows us to understand how mass is distributed along objects that have significant length in comparison to their width or thickness.
This simplification is important because it allows us to describe small elements of the wire in terms of their arc length. Suppose you have a small segment of the wire with an infinitesimally small arc length \( ds \). The mass of this small segment can be expressed as \( dm = \lambda \, ds \). This makes it much more manageable to calculate forces acting on or by the wire, by integrating over these small elements.
In the context of the semicircular wire problem, understanding linear mass density is critical to solve for the gravitational force exerted by the wire on a point mass at the center of its curvature.

Role of Symmetry in Physics

Symmetry significantly simplifies problems in physics by reducing the complexity of calculations. In the case of the semicircular wire, the problem is symmetric around the wire's central vertical axis. This symmetry means that any forces in the horizontal direction stemming from opposing segments of the wire will cancel each other out.
Essentially, for every small mass element on one side of the semicircle contributing a horizontal force component, there is a corresponding element on the opposite side that contributes an equal and opposite component. Because of this, we can focus only on the vertical components when calculating the net gravitational force on the point mass. This is a prime example of how symmetry helps simplify problem-solving by narrowing down the components we need to consider.
Symmetry in physics is not just a mathematical convenience but also displays the proportional behavior of physical systems, making it a powerful tool in solving various physics problems.

Integrating Force Components

Integration is a mathematical tool used to sum up small quantities over a continuous range, and it plays a crucial role in calculating force components in physics problems. In the problem of finding the gravitational force exerted by the semicircular wire, integration is key to summing the infinitesimal gravitational forces \( dF_y \) along the wire.
The vertical component \( dF_y \) of the force due to a small segment of the wire is expressed as \( dF_y = \frac{G \, m \, \lambda \, ds}{R^2} \sin \theta \), where \( \theta \) is the angle formed between the segment and the vertical. Since the wire is semicircular, \( \theta \) ranges from \( 0 \) to \( \pi \).
The integral to find the summed vertical force \( F_y \) is \[ F_y = \int_0^\pi \frac{G \, m \, \lambda}{R^2} \sin \theta \, R \, d\theta \], which simplifies to \( F_y = \frac{2 \, G \, m \, M}{R^2 \pi} \) upon solving. This integration process thus transforms the problem of finding a net force from considering tiny forces across complex shapes into a manageable calculation.