Question

Four forces do the following amounts of work and produce the indicated powers: $$ \begin{array}{|c|c|c|} \hline \text { Force } & \text { Work } & \text { Power } \\ \hline \text { A } & 40\rfloor & 80 W \\ \hline \text { B } & 35\rfloor & 5 W \\ \hline \text { C } & 75\rfloor & 25 \mathrm{~W} \\ \hline \text { D } & 60\rfloor & 30 \mathrm{~W} \\ \hline \end{array} $$ Rank these forces in order of increasing time required to do the work. Indicate ties where appropriate.

Short answer

A, D, C, B

Step-by-step solution

Understand the Relationship Between Work, Power, and Time

Power (P) is defined as the rate of doing work. The formula is \( P = \frac{W}{t} \), where W is work and t is time. The time taken for work to be done can be calculated by rearranging the formula to \( t = \frac{W}{P} \).

Calculate Time for Each Force

Using the formula \( t = \frac{W}{P} \), calculate the time for each force.- For Force A: \( t_A = \frac{40}{80} = 0.5 \) seconds- For Force B: \( t_B = \frac{35}{5} = 7 \) seconds- For Force C: \( t_C = \frac{75}{25} = 3 \) seconds- For Force D: \( t_D = \frac{60}{30} = 2 \) seconds

Rank the Forces Based on Time

List the forces in order of increasing time: - Force A: 0.5 seconds - Force D: 2 seconds - Force C: 3 seconds - Force B: 7 seconds Thus, the order of increasing time is A, D, C, B.

Key concepts

Calculation of Power

Power is a measure of how quickly work is done. It tells us how fast energy is being used or transferred. Mathematically, power is defined by the formula:\[ P = \frac{W}{t} \]where:

• \( P \) is power, measured in watts (W)
• \( W \) is work, measured in joules (J)
• \( t \) is time, measured in seconds (s)

A higher power indicates that more work is done in a shorter amount of time, while a lower power means that the work takes longer. By calculating power using the formula, you can easily compare how fast different tasks are completed.

Time Determination

Calculating the time required to perform a certain amount of work is an essential part of understanding the work-energy relationship in physics. From the equation for power \( P = \frac{W}{t} \), we can derive a formula to determine time \( t \) as follows:\[ t = \frac{W}{P} \]
This rearrangement allows us to calculate exactly how long a force takes to perform a specific amount of work when power is known.

• For example, if a force does 40 joules of work with a power of 80 watts, the time taken would be:\( t = \frac{40}{80} = 0.5 \) seconds.
• Similarly, a force doing 35 joules of work at a power of 5 watts would require \( t = \frac{35}{5} = 7 \) seconds.

Knowing the time helps you compare and rank various forces or machines based on their efficiency and speed.

Relationship Between Work and Time

The relationship between work and time informs us about the efficiency of energy usage. Work, expressed in joules, is a measure of energy transfer, and when combined with time, it allows us to calculate power.
Understanding this relationship is key to solving problems about energy efficiency and time management in physics. When you can determine how long it takes for work to be done at a certain power level, you can analyze and optimize various processes:

• Faster completion of work is achieved with higher power levels, as it reduces the time needed.
• Slower tasks may indicate less power, requiring more time to achieve the same work output.

Once you comprehend this connection, you can apply it to real-life scenarios, such as calculating how much time a motor needs to move an elevator or how quickly a computer processes tasks with varying workloads.