Question

Bansi does a given amount of work in \(30 \mathrm{sec}\). Jaimeen does the same amount of work .. in \(15 \mathrm{sec}\). The ratio of the output power of Bansi to the Jaimeen is.... (A) \(1: 1\) (B) \(1: 2\) (C) \(2: 1\) (D) \(5: 3\)

Short answer

The ratio of the output power of Bansi to Jaimeen is \(1:2\). The answer is (B) \(1: 2\).

Step-by-step solution

Given information

We know that Bansi complete a given amount of work in 30 seconds and Jaimeen complete the same amount of work in 15 seconds. For Bansi: \(Time_1 = 30s\) For Jaimeen: \(Time_2 = 15s\)

Compute the output power of Bansi and Jaimeen

Using the equation for power, \(Power = \dfrac{Work}{Time}\), we will calculate the output power for both Bansi and Jaimeen. Let's consider the amount of work done is W. For Bansi: \(Power_1 = \dfrac{W}{Time_1}\) For Jaimeen: \(Power_2 = \dfrac{W}{Time_2}\)

Calculate the ratio of the output power

Now we will find the ratio of the output power of Bansi to Jaimeen. \(Ratio = \dfrac{Power_1}{Power_2}\) \(Ratio = \dfrac{\dfrac{W}{Time_1}}{\dfrac{W}{Time_2}}\) As the amount of work (W) is the same for both Bansi and Jaimeen, we can cancel W in the equation: \(Ratio = \dfrac{Time_2}{Time_1}\)

Substitute Given Values

Substitute the given values of time for Bansi (30 seconds) and Jaimeen (15 seconds) into the ratio formula: \(Ratio = \dfrac{15}{30}\) Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (15): \(Ratio = \dfrac{1}{2}\)

Write the ratio in the form \(x: y\)

To convert the ratio into \(x:y\) form, we can simply take the reciprocals of both numbers: Ratio of the output power of Bansi to Jaimeen: \(1:2\) The answer is (B) \(1: 2\).

Key concepts

Work and Energy

In physics, work and energy are fundamental concepts that help us understand how forces affect movement within systems. **Work** is defined as the energy transferred to or from an object via the application of force along a displacement. Simply put, work is done when a force causes an object to move. The equation for work is given by:

• \( \text{Work} = \text{Force} \times \text{Displacement} \times \cos(\theta) \)

where \( \theta \) is the angle between the force and the displacement direction.
**Energy** is the capacity to do work. It can exist in various forms, such as kinetic energy, potential energy, thermal energy, and more. In many physical processes, energy is conserved, meaning it can change forms but not be created or destroyed. This conservation principle is crucial in solving problems related to work and energy.
Understanding these two interconnected concepts allows us to analyze various situations, such as lifting an object off the ground, compressing a spring, or, as in our original exercise, comparing the work done by two different people within a certain timeframe. They help in determining how much effort or energy input is required to accomplish specific tasks.

Power equation

Power in physics is a measure of the rate at which work is done or energy is transferred over time. It provides insight into how quickly or efficiently work can be completed within a given time period. The formula for power is expressed as:

• \( \text{Power} = \frac{\text{Work}}{\text{Time}} \)

This equation reveals that power increases as the amount of work done increases or the amount of time taken decreases.
In the context of the original exercise, understanding this equation helped determine the power outputs of two individuals, Bansi and Jaimeen, as they performed the same work in different time intervals. Bansi and Jaimeen's respective power outputs were calculated by dividing the amount of work \( W \) by the time it took each person to complete the work.
This concept of power is pivotal in fields ranging from engineering to athletics, where analyzing efficiency or performance over time is essential. By knowing how to use the power equation, we can assess not only how much work is done but also how fast or efficiently it is completed.

Time and Work relationship

The relationship between time and work is direct and significant in solving many physics problems involving power and energy. When discussing power output, it's crucial to understand this relationship because it affects how we interpret performance and efficiency.
In the formula \( \text{Power} = \frac{\text{Work}}{\text{Time}} \), keeping the amount of work constant, the variation in time directly influences power:

• Decreasing the time taken to perform a task while keeping the work constant results in increased power.
• Conversely, increasing the time decreases the power if the work remains unchanged.

In the original exercise, this relationship was illustrated by Bansi and Jaimeen completing the same work. Jaimeen took 15 seconds, whereas Bansi took 30 seconds. By applying the power equation, we observed that Jaimeen's power output was greater since he completed the work in a shorter time.
This concept is vital in various real-world applications, such as optimizing workflows, designing efficient machines, and ensuring that energy expenditures are minimized while maximizing productivity. Understanding how time influences work and power can lead to more effective strategies for achieving goals within constraints.