Question

4.40 A store sign of mass \(4.25 \mathrm{~kg}\) is hung by two wires that each make an angle of \(\theta=42.4^{\circ}\) with the ceiling. What is the tension in each wire?

Short answer

Answer: The tension in each wire is approximately 31.13 N.

Step-by-step solution

Draw a Free Body Diagram

Draw a Free Body Diagram showing the forces acting on the sign. We have the gravitational force acting downward (weight of the sign) and the tension forces acting along the wires.

Identify the Forces

In this case, we have three forces: The weight W of the sign acting downward and the tension T in each wire acting at an angle of 42.4 degrees with the ceiling (horizontal). We can write the weight as: W = mg = 4.25 * 9.8 N

Break the Tensions into Components

The tension force in each wire can be broken down into horizontal and vertical components. For each wire, the horizontal component is Tx = T*cos(42.4) and the vertical component is Ty = T*sin(42.4).

Equilibrium Condition (Horizontal Direction)

In the horizontal direction, the net force should be zero. As there are two wires with the same horizontal tension component, we can write this as: 2 * T * cos(42.4) = 0 As the tension must be non-zero, this indicates that the two horizontal components must be equal and opposite, meaning: T*cos(42.4) = T*cos(42.4)

Equilibrium Condition (Vertical Direction)

In the vertical direction, the net force should also be zero. The sum of the vertical tension components should equal the weight of the sign. We can write this as: 2 * T * sin(42.4) = W Substitute the value of W into this expression: 2 * T * sin(42.4) = 4.25 * 9.8

Solve for Tension T

Now, we can solve for T by dividing both sides of the equation by 2 and sin(42.4): T = (4.25 * 9.8) / (2 * sin(42.4)) Calculate the value of tension T: T ≈ 31.13 N Thus, the tension in each wire is approximately 31.13 N.

Key concepts

Free Body Diagram

When solving physics problems involving forces, the first step is typically to visualize all the forces acting upon an object using a free body diagram (FBD). An FBD is a simplified representation of an object, often a dot or a box, with vector arrows showing the forces acting on it. Each force is labeled appropriately, and their directions indicate where the force is applied.
In the context of the store sign problem, the FBD illustrates the gravitational force, acting directly downward, and the two tension forces, which are directed along the wires and at an angle to the ceiling. This helps us conveniently analyze each force separately and is crucial for understanding how the object interacts with its surroundings.

Equilibrium Conditions

The concept of equilibrium is central to many physics problems, especially when forces are involved. An object is in mechanical equilibrium when the total forces acting upon it are balanced, meaning there is no net force causing acceleration in any direction. This can be described using Newton's first law of motion, often stated as 'an object in motion stays in motion, and an object at rest stays at rest unless acted upon by a net external force.'
For the store sign to remain stationary, it must satisfy two equilibrium conditions: one for the vertical direction and one for the horizontal direction. These conditions dictate that the collective force of all upward or downward forces, as well as leftward or rightward forces, must equal zero. Hence, we analytically establish a system of equations representing these equilibrium conditions to solve for unknown variables, like the tension in the wires.

Forces in Physics

Forces are one of the fundamental concepts in physics, representing interactions that can cause an object to accelerate. They are vector quantities, possessing both magnitude and direction, and can be categorized in various ways, such as contact forces (e.g., friction, tension) or field forces (e.g., gravity).
In our exercise, we encounter two types of forces: gravitational force, which is the weight of the sign pulling it downward towards the Earth and is a field force, and tension force, which is a contact force transmitted through the wires the sign hangs from. Understanding the nature of these forces allows us to apply the right principles and equations to find unknown quantities like the magnitude of the tension in the wires.

Vector Components

Vectors are mathematical objects used to represent quantities with both magnitude and direction, such as forces. Often, it's necessary to break down a vector into its components along perpendicular axes (usually horizontal and vertical in physics problems) to simplify analysis and calculation.
In the case of the tension in the store sign, the force exerted by each wire is not purely vertical or horizontal but at an angle. By using trigonometric functions (sine and cosine), we can find the horizontal (Tx) and vertical (Ty) components of the tension. This process is crucial when dealing with forces at angles because it allows us to apply equilibrium conditions to each axis independently, ultimately helping us to solve for the tension in the wires.