Question

Two uniform planks, each of mass \(m\) and length \(L,\) are connected by a hinge at the top and by a chain of negligible mass attached at their centers, as shown in the figure. The assembly will stand upright, in the shape of an \(A,\) on a frictionless surface without collapsing. As a function of the length of the chain, find each of the following: a) the tension in the chain, b) the force on the hinge of each plank, and c) the force of the ground on each plank.

Short answer

Based on the given problem and step-by-step solution, create a short answer question: Question: A teacher set up two identical planks on a hinge, and a single chain was attached to both planks with an equal length. Knowing the length of the chain, the mass of the planks, and the distance between the hinges, calculate the tension in the chain, the force on the hinge of each plank, and the force of the ground on each plank. Answer: To calculate the required forces, first identify all the forces acting on the planks and analyze them in the vertical and horizontal directions. Then, solve the equilibrium equations for forces perpendicular and parallel to the hinge to find the tension in the chain (T), the force on the hinge of each plank (F_H), and the force of the ground on each plank (F_v). The expressions for each force are as follows: a) Tension in the chain: \(T = \frac{mg}{\sqrt{1+\frac{L^2}{4d^2}}}\) b) Force on the hinge of each plank: \(F_H = \sqrt{F_h^2 + F_{h_v}^2}\) c) Force of the ground on each plank: \(F_v = F_h + T\frac{L}{2d}\)

Step-by-step solution

Step 1:Identify forces on the planks

We have the following forces acting on the planks: 1. The gravitational force acting at the center of the mass of each plank, \(mg\). 2. The tension in the chain acting at the center of each plank, \(T\). 3. The horizontal force exerted on each plank by the hinge, \(F_h\). 4. The vertical force exerted on each plank by the ground, \(F_v\). 5. The vertical force exerted on each plank by the hinge, \(F_{h_v}.\)

Examine forces acting perpendicular to the hinge

Let the angle between the plank and the vertical axis be denoted by \(\theta\). We'll analyze the forces acting perpendicular to the hinge point (\(F_v\) and the horizontal components of the tension and hinge force) and set up the equilibrium equation: \(\sum F_{perpendicular} = 0 = F_v - F_h - T\sin{\theta}\)

Examine forces acting parallel to the hinge

Now, we'll analyze forces acting parallel to the hinge point (\(F_{h_v}\) and the vertical components of the tension and gravitational force) and set up the equilibrium equation: \(\sum F_{parallel} = 0 = F_{h_v} - T\cos{\theta} + mg\)

Calculate the tension in the chain

Now, we will find the tension in the chain (\(T\)) using the vertical and horizontal components of the equilibrium equation: \(T\sin{\theta} = F_v - F_h\) \(T\cos{\theta} = mg - F_{h_v}\) Divide the two equations above: \(\tan{\theta} = \frac{F_v - F_h}{mg - F_{h_v}}\) We know that \(\tan{\theta}=\frac{L}{2d}\), where \(d\) is half the length of the chain. Therefore: \(T = \frac{mg}{cos{\theta}} = \frac{mg}{\sqrt{1+\frac{L^2}{4d^2}}}\)

Find the force on the hinge of each plank

From the equilibrium equations for forces perpendicular and parallel to the hinge: \(F_h = F_v - T\sin{\theta} = F_v - T\frac{L}{2d}\) \(F_{h_v} = mg - T\cos{\theta} = mg - T\frac{d}{\sqrt{L^2+4d^2}}\) We can find the magnitude of the total force on the hinge of each plank using the Pythagorean theorem: \(F_H = \sqrt{F_h^2 + F_{h_v}^2}\)

Find the force of the ground on each plank

From the equilibrium equation for forces perpendicular to the hinge: \(F_v = F_h + T\sin{\theta} = F_h + T\frac{L}{2d}\) So, we have found the tension in the chain, the force on the hinge of each plank, and the force of the ground on each plank: a) Tension in the chain: \(T = \frac{mg}{\sqrt{1+\frac{L^2}{4d^2}}}\) b) Force on the hinge of each plank: \(F_H = \sqrt{F_h^2 + F_{h_v}^2}\) c) Force of the ground on each plank: \(F_v = F_h + T\frac{L}{2d}\)

Key concepts

Tension in Physics

Tension is a force transmitted through a string, cable, or chain. It is a pulling force and always along the length of the object transmitting it. In our problem, the chain connecting the two planks is under tension. This force is crucial as it prevents the system from collapsing into the frictionless ground. Tension can be calculated using the components of forces acting on the system. By understanding the balance of these forces, we can derive the expression for tension. In the given problem, tension in the chain was calculated based on the angle of the planks and the gravitational forces acting upon them. The tension maintains equilibrium by holding the planks at a fixed angle, effectively ensuring the system remains stable. This stability arises because tension counteracts the forces trying to bring the planks down, allowing them to form the desired shape, in this case, an 'A'. Moreover, understanding tension is vital for solving problems involving chains, strings, or cables in physics, as it allows us to predict how systems will behave under different conditions.

Forces in Mechanics

Forces are fundamental in mechanics as they describe interactions that can change the motion of objects. In the given exercise, several forces are at play.

• The gravitational force on each plank, which acts vertically downward at the center of the plank.
• The tension in the chain, which exerts a pulling force toward the chain's center.
• Horizontal and vertical forces exerted by the hinge and ground.

These forces ensure the system remains in static equilibrium. In simpler terms, static equilibrium is a state where all forces cancel each other out, leading to no net force or movement on the object. A practical way to determine whether a system is in equilibrium is to use two conditions. First, the sum of the forces acting on the system must be zero. This means, the forces in the horizontal direction must balance out with forces in the opposite direction. Second, the sum of moments (or torques) about any axis must also be zero for rotational equilibrium. These principles ensure that the planks stay upright and stable. Understanding these forces gives insight into how different components interact within a mechanical system and allows us to predict how systems will behave under various conditions.

Physics Problem Solving

Physics problem solving often involves breaking down complex systems into simpler parts and analyzing the interactions between these parts. For the assembly of two planks connected by a chain, problem solving includes several key steps:

• Identifying all forces acting on the system. In this case, these include the gravitational force, tension in the chain, and forces at the hinge and from the ground.
• Applying the equilibrium conditions to solve for unknowns like tension and forces at the hinge.
• Using trigonometry to relate angles and lengths, which helps in deriving a practical expression for tension, represented by the angle formed by the planks.

Equations derived from these analyses represent the balance of forces and help in determining the system’s stability. Simplifying these equations often involves using algebraic manipulations and understanding basic trigonometric identities. Problem solving in physics requires a strategic approach that combines physical intuition with mathematical techniques to arrive at a coherent solution. Understanding this process can significantly enhance one’s ability to tackle similar physics problems, as it builds a framework for analyzing the forces and motions in any mechanical system.