The cable of a crane is lifting a 750 kg girder. The girder increases its speed from 0.25 m/s to 0.75 m/s in a distance of 3.5 m. How much work is done by gravity?

검증된 단계별 안내

Step 1: Recall the formula for work done by gravity, which is given by \( W = F \cdot d \cdot \cos(\theta) \), where \( F \) is the gravitational force, \( d \) is the displacement, and \( \theta \) is the angle between the force and displacement vectors.

Step 2: Calculate the gravitational force \( F \) acting on the girder. The gravitational force is given by \( F = m \cdot g \), where \( m \) is the mass of the girder (750 kg) and \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)).

Step 3: Determine the angle \( \theta \) between the gravitational force and the displacement. Since the girder is being lifted vertically, the gravitational force acts downward, while the displacement is upward. Thus, \( \theta = 180^\circ \), and \( \cos(180^\circ) = -1 \).

Step 4: Substitute the values into the work formula. Using \( F = m \cdot g \), \( d = 3.5 \, \text{m} \), and \( \cos(\theta) = -1 \), the work done by gravity is \( W = (750 \cdot 9.8) \cdot 3.5 \cdot (-1) \).

Step 5: Simplify the expression to find the work done by gravity. Note that the negative sign indicates that gravity does work in the direction opposite to the displacement.

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주요 개념

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

Work Done by Gravity

Work done by gravity is calculated using the formula W = mgh, where W is the work, m is the mass, g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the height through which the object moves. In this context, the work done by gravity will be negative if the object is moving upward, as gravity opposes the motion.

Kinetic Energy

Kinetic energy (KE) is the energy an object possesses due to its motion, given by the formula KE = 0.5mv², where m is the mass and v is the velocity. Understanding kinetic energy is essential for analyzing the changes in energy as the girder accelerates from 0.25 m/s to 0.75 m/s, which will help in determining the net work done on the girder.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the work done by the crane must equal the change in kinetic energy of the girder plus the work done against gravity, allowing for a comprehensive analysis of the forces and energy involved in lifting the girder.