Problem 84 - College Physics

Question

To haul a boat out of the water for the winter, a worker at the storage facility uses a wide strap with cables operating at the same angle (measured from the horizontal) on either side of the boat (vFig. 4.48 ). (a) As the boat comes up vertically and \(\theta\) decreases, the tension in the cables (1) increases, (2) decreases, (3) stays the same. (b) Determine the tension in each cable if the boat has a mass of \(500 \mathrm{~kg}\) and the angle of each cable is \(45^{\circ}\)

Step-by-Step Solution

Answer

(a) The tension increases; (b) The tension in each cable is approximately 3464 N.

Step 1: Understanding the Concept

When the angle \( \theta \) decreases, the vertical component of the tension must equal the gravitational force acting on the boat. As \( \theta \) decreases (angle closer to horizontal), to maintain the same vertical force, the tension has to increase.

Step 2: Analyzing Option (a)

As the angle \( \theta \) decreases, the boat is lifted more vertically. To achieve the same net vertical component, the tension in each cable must increase. So, the tension in the cables (1) increases.

Step 3: Calculating the Weight of the Boat

The gravitational force on the boat is given by its weight, \( W = mg \), where \( m = 500 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \). Thus, \( W = 500 \times 9.8 = 4900 \, \text{N} \).

Step 4: Setting Up Tension Equations

For the tension in the cables to balance the weight of the boat, the sum of the vertical components of the tension forces in both cables must equal the weight of the boat: \( 2T \cdot \sin(\theta) = 4900 \, \text{N} \).

Step 5: Solving for Tension in Each Cable

Given \( \theta = 45^{\circ} \), we have \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \). Substituting, we solve for \( T \): \[ 2T \cdot \frac{\sqrt{2}}{2} = 4900 \] simplifies to \( T = \frac{4900}{\sqrt{2}} \approx 3464.10 \, \text{N} \).

Key Concepts

Tension in Cables

Understanding tension in cables involves recognizing how forces interact when pulling or lifting an object. In a typical setup like hoisting a boat, cables are used to distribute the weight across different angles. Here's how it works:

• The tension is the pulling force exerted by each cable to support the object.
• When an object is lifted vertically, this tension must counteract the object's weight.
• The tension force increases as the cables become more horizontal, needing more force to provide the same vertical lift.

When the angle \( \theta \) is larger (closer to horizontal), each cable has to bear more tension. This is why, as angles decrease, tension increases.To calculate the needed tension, break down the forces into components and calculate accordingly. By ensuring the vertical components add up to the weight of the object, you can find the required tension in each cable.

Gravitational Force

Gravitational force is a fundamental force that pulls objects toward each other. On Earth, it gives objects weight, which is the force you feel whenever you lift something. Here's what you need to know:

• Gravitational force on an object at Earth's surface is calculated using the formula \( W = mg \), where \( m \) is the mass and \( g \) is the acceleration due to gravity, approximately 9.8 m/s².
• In the boat problem, the boat's gravitational force (or weight) is determined by multiplying its mass (500 kg) by 9.8 m/s², resulting in 4900 N.
• This force is what the tension in the cables must counteract to lift the boat.

Gravitational force is always directed downward and is crucial in calculating how much force a system must exert to perform activities like lifting or holding an object in place.

Trigonometry in Physics

Trigonometry plays a crucial role in physics, particularly when calculating forces in different directions. This is particularly useful in problems involving angles, like our boat-lifting scenario. Here's why:

• Trigonometry allows the breakdown of forces into components. In this case, the vertical and horizontal components are based on the angle of the cables.
• For angles like 45°, the trigonometric function sine helps find the vertical component of the tension: \( T \sin(\theta) \).
• Given that \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \), you can solve for the tension needed in terms of the known weight of the boat.

Trigonometric principles simplify complex force interactions, making it easier to balance forces and calculate necessary conditions for equilibrium. Understanding these principles is essential for interpreting and solving many physical problems involving angles and forces.