Problem 42 - Trigonometry

Question

Force A chair lift at a ski resort is stopped halfway between two poles that support the cable to which the chair is attached. The poles are 215 feet apart and the combined weight of the chair and the three people on the chair is 725 pounds. If the weight of the chair and the people riding it causes the chair to move to a position \(15.8\) feet below the horizontal line that connects the top of the two poles, find the tension in the cable toward each end of the cable.

Step-by-Step Solution

Answer

The tension in each end of the cable is approximately 2500 pounds.

Step 1: Analyze the Situation

We have a chair lift 15.8 feet below the horizontal line between two support poles that are 215 feet apart. The chair and people have a total weight of 725 pounds. We need to find the tension in each segment of the cable resulting from this setup.

Step 2: Set Up the Free Body Diagram

Consider the chair lift as a point mass located halfway between the poles. We can assume forces are acting vertically downward due to gravitational force (725 pounds) and horizontally as tensions from both directions in the cable.

Step 3: Calculate Horizontal and Vertical Components

Let the tension in the cable be denoted by the variable \( T \). Since the system is symmetrical, the horizontal components of tension on both sides must be equal. Vertically, the cable holds the weight, hence:\[ T \, \sin(\theta) = \frac{725}{2} \]where \( \theta \) is the angle of the cable with the horizontal.

Step 4: Determine the Angle \( \theta \)

Use trigonometry to find \( \theta \). Since the height from the top to the chair is 15.8 feet and the horizontal length of half the span between poles is \( 215/2 = 107.5 \) feet, we have:\[ \tan(\theta) = \frac{15.8}{107.5} \]\[ \theta = \tan^{-1}\left(\frac{15.8}{107.5}\right) \approx 8.34^\circ \]

Step 5: Solve for Tension \( T \)

We use the relationship of vertical components to find the tension. The vertical tension component equates to the vertical force, so using \( \sin(\theta) \):\[ T \, \sin(\theta) = \frac{725}{2} \]Substitute \( \sin(\theta) \approx 0.145 \) from trigonometry:\[ T \, (0.145) = 362.5 \]\[ T = \frac{362.5}{0.145} \approx 2500 \text{ pounds} \]

Step 6: Verify and Conclude

By symmetry and considering equilibrium in horizontal components, the calculated tension on each side is 2500 pounds. Hence, the solution is verified by ensuring the sum of all forces is balanced.

Key Concepts

Free Body Diagram

A free body diagram is a simple illustration used in physics to show all the forces acting upon an object. Imagine sketching the chair lift as a point. This point represents the entire chair and its occupants. Next, draw arrows to indicate forces at play. These arrows show both direction and type of force.

In this ski lift problem, two forces act on the point: the gravitational force and the tension forces from the cable. The gravitational force pulls downward with a magnitude of 725 pounds. This is due to the weight of the chair and people. Two tension forces act along the cable's direction. One tension force pulls towards one pole, and the other tension force pulls toward the opposite pole.

• Gravitational force: 725 pounds, vertically downward
• Tension force (T): horizontally, equal on each side

Free body diagrams help visualize force interactions and are crucial for identifying how forces balance each other out, especially in equilibrium situations.

Tension Calculation

Calculating tension in a cable often involves determining how the cable supports both vertical and horizontal components of force.

In this scenario, we note that the combined weight of the chair and its passengers generates a force of 725 pounds vertically downwards. But the cable's tension works through two components: horizontal and vertical. Given the symmetry of the situation - the chair lift is equidistant from both poles - each segment of the cable supports half the weight. Thus, the total vertical support from one side is 362.5 pounds.

To find tension in the cable, we rely heavily on the vertical component, represented mathematically as:

• Vertical tension component: \ T \sin(\theta) = \frac{725}{2} = 362.5 \

From this, we solve the expression for tension (\(T\)), knowing \(\sin(\theta)\) can be calculated from trigonometry. Solving for \(T\) helps ensure that we correctly balance the forces and maintain equilibrium in the system.

Trigonometric Functions

Trigonometry plays a vital role in analyzing forces in physics problems, especially those involving angles. The ski lift problem requires calculating angles and forces using trigonometric functions.

To find the angle \(\theta\) at which the cable hangs, we use the tangent function. This is because we have a right triangle where the opposite side (the vertical drop) is known, and we want the angle calculation relative to the horizontal. The formula for this is:

• \ \tan(\theta) = \frac{15.8}{107.5} \

Solving this gives the angle \(\theta\). Knowing \(\theta\), we can use the sine function to find the vertical component of tension:

• \ \sin(\theta) \approx 0.145 \

This value helps calculate the exact tension in the cable. Trigonometric functions allow us to break down forces into vertical and horizontal components and are instrumental in solving static equilibrium problems.