Problem 16 - Applied Mathematics

Question

A particle of weight \(W\) is attached to a point \(C\) of an unstretched elastic string \(\mathrm{AB}\), where \(\mathrm{AC}=4 a / 3, \mathrm{CB}=4 a / 7\). The ends \(\mathrm{A}\) and \(\mathrm{B}\) are then attached to the extremities of a horizontal diameter of a fixed hemispherical bowl of radius \(a\). and the particle rests on the smooth inner surface, the angle BAC being \(30^{\circ}\). Show that the modulus of elasticity of the string is \(W\) and determine the reaction of the bowl on the particle. (U of L)

Step-by-Step Solution

Answer

The modulus of elasticity of the string is W. The reaction of the bowl on the particle balances the forces due to string tension and the weight.

Step 1: Define the Problem

Visualize the problem by considering the elastic string AB stretched across the diameter of a hemisphere. Point C is where the weight W is attached, forming angles with points A and B.

Step 2: Calculate Required Lengths

First, identify and calculate the lengths of the sections AC and CB:Given \(AC = \frac{4a}{3} \)\(CB = \frac{4a}{7} \)

Step 3: Determine Angle BAC Position

Calculating the angles needed to locate point C precisely, including the fact that the angle BAC is given to be 30°.

Step 4: Find the Stretch in the String

Calculate the stretch in the segments AC and CB using the formula for the extension of an elastic string: \(\text{extension} = \text{Final length} - \text{Original length}.\)

Step 5: Apply Hooke’s Law

Using Hooke’s Law to find the force exerted by the string, noting that Hooke's law states:\(F = kx\) where k is the modulus of elasticity and x is the extension.

Step 6: Setup Force Equilibrium

Ensuring the forces acting on the particle are in equilibrium, considering gravity and the tension in the string.

Step 7: Solve for the Modulus of Elasticity

Given the equilibrium conditions, solve for the modulus of elasticity of the string (k).

Step 8: Determine the Reaction Force

Finally, determine the reaction force exerted by the bowl on the particle using the resolved forces from the string tension and weight.

Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle in physics that describes the behavior of elastic materials. It states that the force needed to extend or compress an elastic object by some distance (x) is proportional to that distance. Mathematically, this is expressed as: \[ F = kx \]Here, \(F\) represents the force applied, \(k\) is the modulus of elasticity (a constant unique to each material), and \(x\) is the extension or compression.In our problem, Hooke's Law helps us understand how the elastic string stretches when the weight (W) is attached to point C. By knowing the extensions in segments AC and CB, we can calculate the forces exerted by these string segments.

Force Equilibrium

Force equilibrium is a state where all the forces acting on a particle are balanced, meaning the particle remains at rest or moves with a constant velocity. For equilibrium in our problem, the sum of forces in all directions must equal zero.The weight (W) exerts a downward force on the particle at point C. The elastic string AB exerts forces through its stretched segments AC and CB, pulling the particle towards points A and B. We must consider the angles and forces along the horizontal and vertical directions to ensure equilibrium:

• Vertical direction: The tension's vertical components must balance W.
• Horizontal direction: The horizontal components of tensions in AC and CB must balance each other.

By setting up these balance equations, we ensure the particle remains in equilibrium.

Modulus of Elasticity

The modulus of elasticity, denoted as \(k\), is a measure of the stiffness of an elastic material. It quantifies how much force is needed to stretch or compress the material by a certain amount. In our problem, we are required to show that the modulus of elasticity of the string equals the weight \(W\).Using Hooke's Law and the conditions for force equilibrium, we can derive the value of \(k\). We calculate the extensions along the segments AC and CB, then use the equilibrium conditions to solve for \(k\). Through this analysis, we confirm that the modulus of elasticity matches \(W\). This implies the string's stiffness is exactly enough to balance the weight when stretched by specific amounts.

Geometry in Mechanics

In mechanics, geometry plays a crucial role in understanding the behavior of objects and determining forces. Our problem involves geometric relationships in a hemisphere, with a diameter-defined horizontal string and specific angular positions.Knowing the angles, such as BAC being 30°, helps us determine the exact positions and the corresponding forces along segments AC and CB. The geometric constraints lead us to calculate the lengths and extensions in the string segments accurately. These lengths are essential in applying Hooke's Law and ensuring force equilibrium. The combination of geometric and mechanical principles allows us to work through the problem systematically.

Reaction Force

The reaction force is the force exerted by a surface to support the weight of an object resting on it. In this problem, the bowl exerts a reaction force on the particle resting on its smooth inner surface. This force acts perpendicular to the surface at the contact point.To find the reaction force, we use the equilibrium conditions. The combined contributions of the vertical components of the string tensions must balance the weight \(W\). Simultaneously, these components help us determine the exact magnitude and direction of the reaction force provided by the bowl. This reaction force ensures the particle remains at rest and counteracts the weight and string tensions effectively.