A pile driver lifts a 250 kg weight and then lets it fall onto the end of a steel pipe that needs to be driven into the ground. A fall from an initial height of 1.5 m drives the pipe in 35 cm. What is the average force that the weight exerts on the pipe?

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Step 1: Identify the given values. The mass of the weight is \( m = 250 \, \text{kg} \), the initial height is \( h = 1.5 \, \text{m} \), and the distance the pipe is driven into the ground is \( d = 0.35 \, \text{m} \). The goal is to find the average force exerted on the pipe.

Step 2: Calculate the gravitational potential energy of the weight before it falls using the formula \( E_p = m g h \), where \( g \) is the acceleration due to gravity (\( g = 9.8 \, \text{m/s}^2 \)). This energy is converted into work done on the pipe.

Step 3: Relate the work done to the average force exerted on the pipe. The work-energy principle states that \( W = F_{\text{avg}} \cdot d \), where \( W \) is the work done, \( F_{\text{avg}} \) is the average force, and \( d \) is the distance the pipe is driven into the ground.

Step 4: Substitute \( W = E_p \) into the work formula. This gives \( F_{\text{avg}} = \frac{E_p}{d} \). Replace \( E_p \) with \( m g h \) to get \( F_{\text{avg}} = \frac{m g h}{d} \).

Step 5: Plug in the known values \( m = 250 \, \text{kg} \), \( g = 9.8 \, \text{m/s}^2 \), \( h = 1.5 \, \text{m} \), and \( d = 0.35 \, \text{m} \) into the formula \( F_{\text{avg}} = \frac{m g h}{d} \) to calculate the average force exerted on the pipe.

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주요 개념

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

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its height above the ground. It is calculated using the formula GPE = mgh, where m is mass, g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the height. In this scenario, the weight of the pile driver has GPE when lifted, which is converted to kinetic energy as it falls.

Kinetic Energy and Impact Force

Kinetic energy (KE) is the energy of an object in motion, given by the formula KE = 0.5mv², where m is mass and v is velocity. When the weight falls and strikes the pipe, its kinetic energy is transferred to the pipe, causing it to move. The average force exerted on the pipe can be determined by analyzing the change in kinetic energy and the distance over which the force acts.

Newton's Second Law of Motion

Newton's Second Law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). This principle is crucial for calculating the average force exerted by the weight on the pipe, as it relates the net force to the mass of the falling weight and the deceleration experienced by the pipe as it is driven into the ground.