A $750.0$ -kg boulder is raised from a quarry $125$ m deep by a long uniform chain having a mass of $575$ kg. This chain is of uniform strength, but at any point it can support a maximum tension no greater than $2.50$ times its weight without breaking. How long does it take to be lifted out at maximum acceleration if it started from rest?

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Determine the total weight of the boulder and the chain. The weight of the boulder is given by $ W_b = m_b g $, where $ m_b = 750.0 \, \text{kg} $ and $ g = 9.8 \, \text{m/s}^2 $. The weight of the chain is $ W_c = m_c g $, where $ m_c = 575 \, \text{kg} $.

Calculate the maximum tension the chain can support. The chain can support a maximum tension of $ T_{max} = 2.50 \times W_c $. This tension must account for the combined weight of the boulder and the chain, as well as the force required for acceleration.

Set up the force equation for the system. The total force required to lift the boulder and chain with acceleration $ a $ is $ T = (m_b + m_c)(g + a) $. Ensure that $ T \leq T_{max} $ to avoid breaking the chain.

Solve for the maximum acceleration $ a $ that satisfies $ T \leq T_{max} $. Rearrange the inequality $ (m_b + m_c)(g + a) \leq T_{max} $ to isolate $ a $: $ a \leq \frac{T_{max}}{m_b + m_c} - g $.

Use the kinematic equation to find the time required to lift the boulder out of the quarry. The equation $ d = \frac{1}{2} a t^2 $ relates the depth $ d = 125 \, \text{m} $, the acceleration $ a $, and the time $ t $. Solve for $ t $: $ t = \sqrt{\frac{2d}{a}} $.

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주요 개념

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

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle is crucial for understanding how forces affect the motion of the boulder and the chain. In this scenario, the net force will determine the maximum acceleration at which the boulder can be lifted, factoring in both gravitational and tension forces.

Tension in a Rope or Chain

Tension is the force transmitted through a rope or chain when it is pulled tight by forces acting from opposite ends. In this problem, the chain must support the weight of the boulder while also providing the necessary force to accelerate it upwards. The maximum tension that the chain can withstand is a critical factor in determining how quickly the boulder can be lifted without breaking the chain.

Kinematic Equations of Motion

Kinematic equations describe the motion of objects under constant acceleration. These equations relate displacement, initial velocity, final velocity, acceleration, and time. In this case, they will be used to calculate the time it takes for the boulder to be lifted from rest to a certain height under maximum acceleration, providing a mathematical framework to solve the problem.