Problem 5 - University Physics with Modern Physics

Question

A picture frame hung against a wall is suspended by two wires attached to its upper corners. If the two wires make the same angle with the vertical, what must this angle be if the tension in each wire is equal to 0.75 of the weight of the frame? (Ignore any friction between the wall and the picture frame.)

Step-by-Step Solution

Answer

The angle \( \theta \) must be approximately \( 48.19^\circ \).

Step 1: Analyze the Forces

Consider the forces acting on the picture frame. There is the weight of the frame acting downward and the tension in the two wires, each pulling at an angle \( \theta \) from the vertical.

Step 2: Set up the Equations

Let \( W \) be the weight of the frame. The vertical component of the tension in each wire will balance the weight of the picture. If \( T \) is the tension in one wire, and given that \( T = 0.75W \), the sum of the vertical components can be written as: \( 2T \cos \theta = W \).

Step 3: Substitute Known Values

Substituting \( T = 0.75W \) into the equation: \( 2(0.75W) \cos \theta = W \). Simplify this to: \( 1.5W \cos \theta = W \).

Step 4: Solve for \( \cos \theta \)

Divide both sides of the equation by \( W \): \( 1.5 \cos \theta = 1 \). Thus, \( \cos \theta = \frac{1}{1.5} = \frac{2}{3} \).

Step 5: Calculate \( \theta \)

Find \( \theta \) by calculating the inverse cosine: \( \theta = \cos^{-1}(\frac{2}{3}) \).

Step 6: Conclusion

Using a calculator, \( \theta \approx 48.19^\circ \).

Key Concepts

Forces in Equilibrium

In physics, the concept of forces in equilibrium refers to a situation where all acting forces on an object are balanced, resulting in no net force. This means the object remains at rest or continues to move with constant velocity. In the context of the picture frame scenario, the frame is suspended and stationary, implying it is in equilibrium. There are two key factors at play:

• Weight: This is the gravitational force acting downward on the picture frame. It is a result of the mass of the frame multiplied by the acceleration due to gravity.
• Tension: Each wire exerts an upward force to counterbalance the frame's weight, ensuring the net force is zero. Since the system is at rest, the combined upward forces of the wires' tensions equal the downward gravitational force.

This state of balance requires that all vertical components of the forces sum to zero. Therefore, understanding equilibrium helps in setting up the correct equations that describe the system.

Tension in Ropes

Tension is a crucial force when dealing with ropes or wires, especially in situations involving objects being suspended. In our problem, tension refers to the force exerted along the length of each wire that holds the picture frame. Key characteristics of tension include:

• Uniform Tension: Each wire carries the same tension force due to symmetry and equal angles with the vertical.
• Relationship with Weight: The given tension in each wire is 0.75 times the weight of the frame. This relationship is instrumental in calculating the required angle that the wires make with the vertical.
• Direction: The tension force is directed along the rope or wire away from the object it supports, balancing components must be computed to analyze its effect.

Understanding tension is vital as it allows us to determine the necessary conditions to maintain equilibrium, ensuring the frame remains stationary while suspended.

Trigonometry in Physics

Trigonometry is an essential mathematical tool in physics, especially in analyzing problems involving angles and forces. It helps us break down complex vectors into their components, which is crucial for solving equilibrium problems.For the picture frame:

• Components of Force: Each of the tensions in the wires can be divided into vertical and horizontal components using trigonometric functions. The vertical component contributes to balancing the frame's weight.
• Cosine Function: In this problem, we use the cosine function to relate the vertical component of tension to the weight. Solving for \(\theta\) from \(\cos \theta = \frac{2}{3}\) illustrates this application.
• Inverse Trigonometric Functions: Finding the angle \(\theta\) by calculating the inverse cosine allows us to determine the exact angle each wire makes with the vertical, which was around 48.19 degrees.

Thus, trigonometry provides the necessary methods to dissect force vectors, making it integral in physics problem-solving.