Question

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 26.67 N.

Step-by-step solution

Calculate the weight of the store sign

To find the tension in each wire, we need to know the total weight of the store sign. The weight (W) can be calculated using the equation: W = m * g where m is the mass of the store sign (4.25 kg) and g is the gravitational acceleration (approximately \(9.81\mathrm{~m/s^2}\)). Let's calculate the weight of the store sign: W = 4.25 kg * 9.81 m/s² W = 41.6975 N (approx) Now we have the total weight of the sign, which is 41.6975 N.

Find the vertical and horizontal tension components

Next, let's split the tension (T) in each wire into its vertical and horizontal components (T_v and T_h, respectively). Since both T_v components support the weight of the store sign, we can write the equations: 2 * T_v = W And since the store sign is in equilibrium, the horizontal components should cancel each other: T_h = T_h Now, let's use trigonometry to relate T_v and T_h to the total tension T in each wire: T_v = T * sin(θ) T_h = T * cos(θ)

Solve for T

Substitute the equations for T_v and T_h into the equation for the weight: 2 * T * sin(θ) = W Solve for T: T = W / (2 * sin(θ)) Plug in the values for W and θ: T = 41.6975 N / (2 * sin(42.4°)) Now, calculate the tension in each wire: T ≈ 26.67 N (approx) So, the tension in each wire is approximately 26.67 N.

Key concepts

Trigonometry in Physics

Trigonometry plays a pivotal role in physics, particularly when dealing with forces acting at angles. It is the branch of mathematics that deals with the relationships between the sides and angles of triangles. In physics problems, especially those involving tension in wires, understanding trigonometry is essential.

When a wire is angled, the tension within is not solely vertical or horizontal; it’s a combination of both, known as components. The trigonometric functions sine (sin) and cosine (cos) help us decompose the tension into vertical and horizontal components, respectively. For example, with a wire forming an angle \( \theta \) with the ceiling, \( T_v = T \cdot \sin(\theta) \) gives the vertical component responsible for supporting the hanging object’s weight, while \( T_h = T \cdot \cos(\theta) \) represents the horizontal component that often ensures stability.

In the store sign problem, these trigonometric principles are used to find the tension in the wires. The angle given, \( \theta=42.4^\circ \) is crucial for calculating both components, which are needed to ensure the sign remains at equilibrium.

Importance of Accurate Angle Measurement

It's crucial to note that accurate angle measurement is significant. Even a small error in the angle can lead to a substantial difference in the calculation of tension components, which in turn affects the safety and stability of structures or objects being supported.

Equilibrium of Forces

The concept of equilibrium of forces is fundamental in physics problems that involve static objects—that is, objects at rest. Equilibrium occurs when all the forces acting on an object are balanced, resulting in no net force and consequently no acceleration.

In equilibrium, there are two conditions that must be met. The first condition is that all horizontal forces must balance out, and the second is that all vertical forces must also do the same. The wires' tensions and gravitational force are the key players in this scenario. The tension in the wires must provide a vertical force that equals the downward gravitational force, while the horizontal components of the tension should cancel out—since they are equal in magnitude and opposite in direction.

In the problem at hand, once we know the total weight of the sign (which acts directly downwards due to gravity), we can set it against the combined vertical components of the tension in each wire, thus utilizing the equilibrium condition \( 2 \cdot T_v = W \) to solve for the individual tension in the wires.

Understanding Static Equilibrium

A deep understanding of static equilibrium is not just theoretical—it’s practical. Designers and engineers must calculate forces correctly to ensure the stability of structures. In our store sign example, ensuring that the sign remains static demands precision in balancing forces.

Gravitational Force Calculation

Gravitational force calculation is essential in physics problems that deal with objects subjected to Earth’s gravity. This force, commonly referred to as weight, is the result of the gravitational pull exerted on the mass of an object. The formula to calculate this force is \( W = m \cdot g \) where \( W \) is the weight, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity, which is approximately \( 9.81 \mathrm{~m/s^2} \) on Earth.

In every physics problem like the store sign example, accurately calculating the gravitational force lets us determine the load that the wires must bear. Using the correct value of gravity is important; even slight variations in \( g \) due to geographical changes can affect the results, although these are typically negligible for homework problems and most practical applications.

• Determine the object's mass.
• Multiply by Earth's gravity to find weight.
• Use weight in calculating other forces like tension.

The calculation of the gravitational force was the foundational step in solving the store sign problem. It established the weight that both wires must collectively support in equilibrium, giving an essential value for solving the tension in each wire.