Problem 10 - Applied Mathematics

Question

A uniform rod of length \(l\) rests over the rim of a fixed hemispherical bowl of radius \(r\), with one end in contact with the surface of the bowl. If all contacts are smooth and the inclination of the rod to the horizontal is \(\theta\), prove that the value of \(\theta\) is given by the equation \(8 r \cos ^{2} \theta-l \cos \theta-4 r=0\).

Step-by-Step Solution

Answer

The inclination of the rod, \( \theta \), satisfies the equation: \( 8r \cos^2 \theta - l \cos \theta - 4r = 0 \).

Step 1: Understand the Problem

A uniform rod of length \(l\) rests on a hemispherical bowl of radius \(r\). The rod makes an angle \(\theta\) with the horizontal. Calculate the value of \(\theta\) such that the rod is in equilibrium.

Step 2: Identify Relevant Forces

The forces acting on the rod are the weight \(mg\), the normal reaction force at the contact point with the bowl, and the normal reaction force at the end of the rod on the surface.

Step 3: Establish the Geometrical Relationships

For equilibrium, consider the projection of the rod's length along the horizontal. The horizontal distance from the point of contact with the bowl to the rod's other end is given by \( l \cos \theta \).

Step 4: Analyze the Forces and Torque

Considering torque balance about the contact point with the bowl, the condition for equilibrium can be derived by balancing the moments due to the rod's weight and the normal reaction forces.

Step 5: Use Trigonometric Relationships

By examining the geometry, the bowl’s radius projects onto the rod, forming a right triangle. For simplicity, project onto the axis parallel to the radius and the rod. Using these projections, we derive: \[ r \cos \theta + \frac{l \cos 2\theta}{2} = OB \]

Step 6: Reduce the Equation

Combining all these geometric factors and reducing them algebraically, we obtain: \[ 8r \cos^2 \theta - l \cos \theta - 4r = 0 \]

Key Concepts

geometrical relationships in physics

When understanding the equilibrium of physical objects, the first step is to analyze the geometrical relationships involved. Given a rod resting in a hemispherical bowl, the rod’s positioning and inclination play a crucial role.
The rod’s inclination \(\theta\) with the horizontal determines how the length of the rod projects onto different axes. By examining the projection, we know the horizontal distance from the contact point on the bowl to the end of the rod is derived from the rod's length multiplied by the cosine of its angle with the horizontal, hence \(\text{distance} = l \cos \theta\).
Understanding these basic geometric principles helps in visualizing and solving the problem of equilibrium efficiently.

torque balance

To ensure the rod is in equilibrium within the bowl, we need to consider torque balance. Torque is a measure of how much a force acting on an object causes that object to rotate. Balancing torque means ensuring that the clockwise and counterclockwise moments about any pivot point are equal. In this case, we'll consider the contact point with the bowl as the pivot.
Forces acting on the rod include:

• Weight (\(mg\)) of the rod acting downwards at its center of gravity.
• Normal reaction forces at the point of contact with the bowl and the end of the rod.

Using torque balance, we calculate the moments due to these forces about the contact point. The torque balance equation will consider the perpendicular distances from the pivot point to the line of action of these forces.

trigonometric projections

Trigonometry plays a pivotal role in breaking down forces into their respective components along different axes. With the rod inclined at angle \(\theta\), the key trigonometric functions involved are sine and cosine.
For this problem, the horizontal and vertical projections of the rod’s length when positioned at an angle \(\theta\) are given by:

• Horizontal Projection: \( l \cos \theta \)
• Vertical Projection: \( l \sin \theta \)

Furthermore, the radius \( r \) of the bowl makes specific angles with these projections. These trigonometric relationships are fundamental in deriving the final equilibrium condition involving the angles and the physical dimensions of the problem.

force analysis

Analyzing the forces acting on the rod helps us understand how equilibrium is maintained. The forces include:

• The gravitational force (weight) \( mg \), acting downward at the rod's center of gravity.
• The normal reaction force from the contact on the bowl and at the rod’s end.

For the rod to be in equilibrium, the sum of horizontal and vertical forces must be zero. Additionally, the total torque acting on the system must also sum to zero. This involves setting up and solving force equations in both horizontal and vertical directions, ensuring that the vector sum of forces and moments equals zero.

equilibrium conditions

Equilibrium conditions require that both translational and rotational equilibrium are satisfied. Translational equilibrium means that the sum of all external forces acting on the rod is zero in both horizontal and vertical directions. Rotational equilibrium requires that the sum of all torques (moments) around any point is zero.
Given the rod within a hemispherical bowl, to fulfill these conditions:

• The horizontal and vertical components of all forces should balance out.
• The torque calculated around the pivot point (contact point on the bowl) should be zero.

By combining these equilibrium conditions with the geometrical and trigonometric relationships, we derive the equation: \(8 r \cos^{2} \theta - l \cos \theta - 4 r = 0\), satisfying all criteria for the rod to be at rest.