A 27-kg chandelier hangs from a ceiling on a vertical 3.4-m-long wire. What horizontal force would be necessary to displace its position 0.15 m to one side?

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Determine the forces acting on the chandelier. The chandelier is in equilibrium, so the forces include the tension in the wire, the gravitational force acting vertically downward, and the horizontal force applied to displace it.

Model the situation as a right triangle. The vertical side of the triangle is the length of the wire (3.4 m), the horizontal side is the displacement (0.15 m), and the hypotenuse is the new length of the wire under tension.

Calculate the angle θ that the wire makes with the vertical after displacement using the tangent function: \( \tan(\theta) = \frac{\text{horizontal displacement}}{\text{vertical length}} \). Solve for \( \theta \) using \( \theta = \arctan(\frac{0.15}{3.4}) \).

Express the horizontal force \( F_h \) in terms of the tension \( T \) in the wire and the angle \( \theta \). The horizontal force is given by \( F_h = T \sin(\theta) \).

Determine the tension \( T \) in the wire. The vertical component of the tension balances the gravitational force: \( T \cos(\theta) = mg \), where \( m = 27 \ \text{kg} \) and \( g = 9.8 \ \text{m/s}^2 \). Solve for \( T \), then substitute it into the equation for \( F_h \) to find the horizontal force.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Tension in a Wire

Tension is the force exerted along a wire or rope when it is pulled tight by forces acting from opposite ends. In this scenario, the chandelier's weight creates a downward tension in the wire, which must be balanced by any horizontal forces applied to maintain equilibrium.

Equilibrium of Forces

An object is in equilibrium when the net force acting on it is zero. For the chandelier, this means that the vertical forces (weight and tension) and the horizontal forces (applied force) must balance each other. Understanding this concept is crucial to determine the necessary horizontal force to displace the chandelier.

Torque

Torque is the rotational equivalent of linear force and is calculated as the product of the force and the distance from the pivot point (in this case, the ceiling). When the chandelier is displaced, the horizontal force creates a torque about the point where the wire is attached, which must be countered by the tension in the wire to maintain stability.