Problem 75 - Master Resource Book in JEE Main Physics

Question

A one kilowatt motor is used to pump water from a well \(10 \mathrm{~m}\) deep. The quantity of water pumped out per second is nearly (a) \(1 \mathrm{~kg}\) (b) \(10 \mathrm{~kg}\) (c) \(100 \mathrm{~kg}\) (d) \(1000 \mathrm{~kg}\)

Step-by-Step Solution

Answer

(b) 10 kg

Step 1: Understand the Problem

We need to determine how much water a 1 kW motor can pump from a 10 m deep well in one second. We know that 1 kW is equivalent to 1000 watts, and we will use the formula for power to find the quantity of water.

Step 2: Power Formula

The power exerted when lifting a weight is given by the formula: \( P = \frac{W}{t} \), where \( P \) is power in watts, \( W \) is the work done in joules, and \( t \) is time in seconds. Since we are finding the quantity per second, \( t = 1 \) second.

Step 3: Work Done Calculation

To find the work done, we use the formula \( W = mgh \), where \( m \) is the mass of the water in kg, \( g = 9.8 \, \mathrm{m/s^2} \) is the acceleration due to gravity, and \( h = 10 \, \mathrm{m} \) is the height. So, \( W = m \times 9.8 \times 10 = 98m \).

Step 4: Equate Power and Work Done

Since the power is 1000 W, we set up the equation \( 1000 = 98m \). Solving for \( m \) gives the amount of water, \( m = \frac{1000}{98} \approx 10.2 \mathrm{~kg} \) of water per second.

Step 5: Select the Closest Answer

The closest value to the calculated \( 10.2 \mathrm{~kg} \) from the given options is \( 10 \mathrm{~kg} \), so the answer is option (b).

Key Concepts

Work-Energy Principle

The work-energy principle is a fundamental concept in physics that connects the dots between work done on an object and the change in its kinetic energy. When work is done on an object, energy is transferred, which results in altering the object's motion. In simple terms, the principle states that the work done by forces on an object is equal to the change in its kinetic energy.
For example, if you push a box across the floor, the energy you use to push is transferred to the box, and it starts to move. This principle helps in solving various problems in physics, like determining how much energy is needed to lift an object, as seen in this exercise.
The formula is given by:

• Work Done (W) = Change in Kinetic Energy

A practical application of the work-energy principle is seen in the exercise where a motor lifts water from a well. The motor does work against gravity, increasing the water's potential energy, since it is raised to a higher position.

Conservation of Energy

Conservation of energy is a core principle in physics indicating that energy cannot be created or destroyed; it can only change from one form to another. This fundamental rule assures us that the total energy in a closed system remains constant.
This concept plays a vital role in understanding real-world systems, ensuring we can predict how energy moves and changes within these systems. For instance, the chemical energy in fuel converts to mechanical energy in an engine, or the electrical energy in a battery converts to kinetic energy in a moving toy.
In the context of the exercise, conservation of energy implies that the energy from the motor, which is electrical, is fully converted into potential energy as the water is lifted from the well.
In simpler terms, when the motor pumps water:

• The electrical energy is used to do work, lifting the water.
• All the energy involved in the task transforms from one form into another without any loss.

This helps us understand how the motor's power affects the amount of water pumped.

Mechanical Work

Mechanical work is the process by which a force causes an object to move in the direction of the force applied. It is a measure of energy transfer when an object is displaced by the application of a force. The concept is pivotal in physics to understand how machines function, like motors and engines.
The equation for mechanical work is:

• Work (W) = Force (F) × Distance (d)

This simple formula tells us that work depends on two main factors: the magnitude of the force applied, and the distance over which it is applied.
Let's integrate this with the exercise:

• When the motor pumps water, it applies a force to lift the water against gravity.
• Here, the force is due to gravity, calculated as the weight of the water (mass times gravitational acceleration).
• The distance is the height of the well.

The amount of work done by the motor to lift water is calculated to ensure the energy supplied is used efficiently.