Question

Calculate You raise a bucket of water from the bottom of a well that is \(12 \mathrm{~m}\) deep. The mass of the bucket and the water is \(5.00 \mathrm{~kg}\), and it takes \(15 \mathrm{~s}\) to raise the bucket to the top of the well. How much power is required?

Short answer

The power required is 39.24 W.

Step-by-step solution

Identify the Given Values

First, we need to identify and list all the values provided in the problem. The well is \(12\text{ m}\) deep, the mass of the bucket and water is \(5.00\text{ kg}\), and it takes \(15\text{ s}\) to raise the bucket.

Calculate the Force of Gravity

To find the force of gravity acting on the bucket, use the formula \( F = mg \), where \( m = 5.00 \text{ kg} \) and \( g = 9.81 \text{ m/s}^2 \). \[ F = 5.00 \times 9.81 = 49.05 \text{ N} \]

Determine the Work Done

The work done in lifting the bucket is given by the formula \( W = Fd \), where \( F = 49.05 \text{ N} \) and \( d = 12 \text{ m} \). \[ W = 49.05 \times 12 = 588.6 \text{ J} \]

Calculate the Power Required

Power is the rate at which work is done. Use the formula \( P = \frac{W}{t} \) where \( W = 588.6 \text{ J} \) and \( t = 15 \text{ s} \). \[ P = \frac{588.6}{15} = 39.24 \text{ W} \]

Conclusion

The power required to raise the bucket to the top of the well is \(39.24 \text{ W}\).

Key concepts

Work Done

The concept of "work done" in physics revolves around the energy transfer that occurs when a force moves an object over a distance. It’s a straightforward idea, yet fundamental to understanding energy and mechanics.

Work is defined by the formula: \[ W = F \times d \]where:

• \( W \) is the work done measured in joules \( (J) \)
• \( F \) is the force applied in newtons \( (N) \)
• \( d \) is the distance moved in the direction of the force measured in meters \( (m) \)

In our problem, when the bucket is raised, work is done against the force of gravity. Calculating this work tells us how much energy is needed to lift the bucket from the bottom to the top of the well.

Force of Gravity

Gravity is a natural force that pulls objects towards the center of the Earth. Understanding the force of gravity is crucial for calculating work done when lifting objects. The force of gravity acting on an object is given by the formula:\[ F = mg \]where:

• \( F \) is the gravitational force in newtons \( (N) \)
• \( m \) is the mass of the object in kilograms \( (kg) \)
• \( g \) is the acceleration due to gravity \( 9.81 \text{ m/s}^2 \)

In this exercise, the mass of the bucket and water is \(5.00\text{ kg}\), so the force of gravity is calculated as \( F = 5.00 \times 9.81 = 49.05 \text{ N} \).

This force tells us how much gravitational pull is acting downward while lifting the bucket.

Physics Formula

Physics formulas are tools that help us solve problems by relating different physical quantities. In this problem, we use three main formulas:

• Force of Gravity: \( F = mg \)
• Work Done: \( W = Fd \)
• Power: \( P = \frac{W}{t} \)

These formulas interconnect force, work, and power. They reveal how manipulating one variable affects the others. For example, if distance \(d\) increases, work \(W\) will also increase, assuming \(F\) remains constant.

The goal is to understand the relationships within the formulas, enabling us to predict outcomes or calculate unknowns in different physical situations.

Problem Solving Steps

Solving physics problems often involves breaking them down into smaller, manageable parts. Our exercise follows a structured approach:

• Step 1: Identify the Given Values: Listing the known values saves time and organizes thoughts.
• Step 2: Calculate the Force of Gravity: Using \( F = mg \) we find the gravitational pull acting on the bucket.
• Step 3: Determine the Work Done: With \( W = Fd \), calculate the energy needed to lift the bucket.
• Step 4: Calculate the Power Required: Divide the work done by time \( t \), using \( P = \frac{W}{t} \), to find power.
• Step 5: Conclusion: Sum up your results to ensure they answer the question comprehensively.

This methodical process increases accuracy and boosts confidence in problem-solving across various physics scenarios.