Question

If a man has an average useful power output of \(40.0 \mathrm{W}\) what minimum time would it take him to lift fifty \(10.0-\mathrm{kg}\) boxes to a height of \(2.00 \mathrm{m} ?\)

Short answer

The minimum time to lift fifty 10-kg boxes is approximately 4.09 minutes.

Step-by-step solution

Understanding the Problem

To solve this problem, we need to calculate the work done by the man lifting the boxes, and then determine the minimum time required to do this work, given his average useful power output.

Calculate Work for One Box

Calculate the work done to lift one box. The work done is given by the formula: \( \text{Work} = \text{Force} \times \text{Distance} \).The force required to lift one box is equal to the weight of the box, which is given by \( F = mg \), where \( m \) is the mass (\( 10.0 \, \text{kg} \)) and \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).Calculate: \( F = 10.0 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 98.1 \, \text{N} \).Therefore, the work done to lift one box to a height of \( 2.00 \, \text{m} \) is:\( \text{Work} = 98.1 \, \text{N} \times 2.00 \, \text{m} = 196.2 \, \text{J} \).

Calculate Work for Fifty Boxes

The total work done to lift fifty boxes is the work done for one box multiplied by the number of boxes:\( \text{Total Work} = 196.2 \, \text{J/box} \times 50 \, \text{boxes} = 9810 \, \text{J} \).

Calculate Time Required

The time required to do the work is calculated using the formula: \( \text{Power} = \frac{\text{Work}}{\text{Time}} \).Rearrange this to find\( \text{Time} = \frac{\text{Work}}{\text{Power}} \).Substitute the known values:\( \text{Time} = \frac{9810 \, \text{J}}{40.0 \, \text{W}} = 245.25 \, \text{seconds} \).

Convert Time to Minutes

To express the time in minutes, divide the result in seconds by 60:\( \frac{245.25 \, \text{seconds}}{60 \, \text{seconds/minute}} \approx 4.09 \, \text{minutes} \).

Key concepts

Work-Energy Principle

The Work-Energy Principle is a fundamental concept in physics that explains how work and energy are related. In simple terms, this principle states that the work done on an object is equal to the change in its kinetic energy. When you apply a force to move an object over a distance, you are doing work on that object. This work can change the object's speed, and in turn, its kinetic energy.
In the context of lifting boxes, as described in the exercise, the work done by the man results in an increase in the potential energy of the boxes as he lifts them against the force of gravity. By understanding this principle, you can see why calculating work is crucial to understanding how the movement of an object affects its energy state. In essence, by performing work on the boxes, their potential energy increases as their position changes.

Calculating Work

Calculating work is a straightforward process if you know the force applied and the distance over which it is applied. The formula for calculating work is:

• Work = Force × Distance.

To determine the force in our example, you need to consider the weight of the box, which is the product of its mass and the gravitational acceleration:

• Force (F) = Mass (m) × Gravitational Acceleration (g).

Gravitational acceleration on Earth is approximately 9.81 m/s². Thus, for a 10 kg box, the weight is calculated as 98.1 N.
The work done to lift each box can then be found by multiplying this force by the height (distance) the box is lifted. For one box lifted 2 meters:

• Work = 98.1 N × 2 m = 196.2 J (Joules).

If multiple boxes are being lifted, the total work is calculated by multiplying the work done for one box by the total number of boxes.

Units of Power

Understanding the units of power is essential to solving problems involving work and time. Power is defined as the rate at which work is done. It tells you how much work can be done in a specific amount of time.
The unit of power is the Watt (W), which corresponds to one Joule per second (1 W = 1 J/s). This means if a power machine performs 1 Joule of work every second, it has a power output of 1 Watt.
In our exercise, the useful power output of 40.0 Watts means that 40 Joules of work can be done every second. To find the time required for a task, such as lifting boxes, you can rearrange the power formula:

• Power = Work / Time, which becomes, Time = Work / Power.

This rearranged formula helps you calculate the time it takes to complete a task when the work and power output are known, making it a valuable tool in various practical scenarios.