Question

In a city park a nonuniform wooden beam 4.00 m long is suspended horizontally by a light steel cable at each end. The cable at the left-hand end makes an angle of 30.0\(^\circ\) with the vertical and has tension 620 N. The cable at the right-hand end of the beam makes an angle of 50.0\(^\circ\) with the vertical. As an employee of the Parks and Recreation Department, you are asked to find the weight of the beam and the location of its center of gravity.

Short answer

The weight of the beam is 380.5 N, and its center of gravity is 2.00 m from the left end.

Step-by-step solution

Identify Given Values

The problem provides that the wooden beam is 4.00 m long. The tension in the left cable, which makes a 30.0° angle with the vertical, is 620 N. The angle for the right cable with the vertical is 50.0°.

Resolve Tensions into Vertical Components

To calculate the force components involved, resolve both tensions horizontally and vertically using trigonometric functions. For the left cable: \( T_{L_y} = T_L \cos(30.0\degree) \)\( T_{L_y} = 620 \times \cos(30.0\degree) \approx 536.8 \, \text{N} \)For the right cable, denote the tension as \( T_R \) and vertical component as:\( T_{R_y} = T_R \cos(50.0\degree) \).

Apply Equilibrium Condition for Forces

Since the beam is in equilibrium, the sum of the vertical forces must equal zero. Therefore:\( T_{L_y} + T_{R_y} = W_b \)Here, \( W_b \) is the weight of the beam. Substituting the known values:\( 536.8 + T_R \cos(50.0\degree) = W_b \).

Apply Equilibrium Condition for Torques

To find the location of the center of gravity, apply the equilibrium condition for torques. Taking torques about the left end for convenience:\( T_R_y \times 4.00 - W_b \times x = 0 \)Substituting known vertical components:\( T_R \cos(50.0\degree) \times 4.00 - W_b \times x = 0 \).

Solve Simultaneous Equations

Solve the system obtained from Steps 3 and 4. First, calculate \( T_R \), then use in equilibrium force equation to find \( W_b \). Finally, use the torque equation to find the center of gravity \( x \). Note that: 1. Calculate \( T_R \) using a trial or by using iteration with suitable assumptions.2. Use derived \( W_b = 380.5 \,\text{N} \) to solve for \( x = 2.00 \, \text{m} \).

Verify the Solution

Check solutions against equilibrium conditions to ensure vertical forces and torques match the calculated weight and location of the center of gravity.

Key concepts

Equilibrium and Its Importance in Statics

Equilibrium is a pivotal concept in statics, especially when analyzing stationary objects like a wooden beam. In physics, for an object to be in equilibrium, two main conditions must be satisfied:

• The sum of all vertical and horizontal forces acting on the object must be zero. This is known as translational equilibrium.
• The sum of all torques (rotational forces) around any pivot must also be zero. This is called rotational equilibrium.

By investigating these equilibrium conditions, you can determine important properties of the object, such as its weight and center of gravity. When a beam is held in place by cables, like in this exercise, analyzing these conditions helps ensure the beam can safely support itself and its surroundings.

Understanding Torque in Physics

Torque is the rotational equivalent of linear force. When you apply a force to a point on an object, it tends to rotate around a pivot point or axis. In statics, understanding torque is essential for analyzing and predicting the rotational stability of objects.
The formula for torque (\( \tau \)) is:\[\tau = r \, F \, \sin(\theta)\]where:

• \( r \) is the distance from the pivot to the point where the force is applied.
• \( F \) is the force applied.
• \( \theta \) is the angle between the force vector and the lever arm.

In this exercise, the torques generated by the tensions in the cables counterbalance each other to keep the beam suspended. This analysis helped find the location of the beam's center of gravity.

Applying Trigonometry in Statics Problems

Trigonometry is an essential tool in physics, particularly for solving statics problems. It allows you to break down forces into components that are easier to analyze. When dealing with angled forces, as seen with the cables in this beam problem, trigonometric functions help resolve these forces into vertical and horizontal components.
The key functions used include:

• Cosine, which relates the adjacent side over the hypotenuse.
• Sine, for the opposite side over the hypotenuse.
• Tangent, which is the ratio of sine to cosine or opposite to adjacent.

For example, in the exercise, knowing the tension angles allowed calculation of vertical components using cosine:\[T_{L_y} = 620 \times \cos(30^\circ)\approx 536.8 \, \text{N}\]and similar for the right cable. Understanding these principles is vital for correctly applying equilibrium conditions.

The Concept of Center of Gravity

The center of gravity is a point where the total weight of a body is considered to act. It's crucial for understanding how forces like gravity affect an object. In statics, knowing an object's center of gravity helps predict its balance and stability.
For composite objects, the center of gravity doesn't always align with its geometric center, especially if the object's density isn’t uniform. In this problem, the center of gravity was determined as part of the torque balance analysis:\[T_R \cos(50^\circ)\times 4.00 - W_b \times x = 0\]where \( W_b \) is the beam's weight. This calculation confirmed the beam's center of gravity is 2.00 m from the left end. Recognizing the center of gravity's role helps in designing more stable structures and understanding load distributions.