Problem 73 - Master Resource Book in JEE Main Physics

Question

A 10 HP motor pump out water from a well of depth \(20 \mathrm{~m}\) and falls a water tank of volume 22380 litre at a height of \(10 \mathrm{~m}\) from the ground the running time of the motor to fill the empty water tank is \(\left(g=10 \mathrm{~ms}^{-2}\right.\) ) (a) \(5 \mathrm{~min}\) (b) \(10 \mathrm{~min}\) (c) \(15 \mathrm{~min}\) (d) \(20 \mathrm{~min}\)

Step-by-Step Solution

Answer

The running time of the motor is 15 minutes.

Step 1: Calculate the Total Work Done

The motor has to lift the water from a depth of \(20 \text{ m}\) to the water tank which is at \(10 \text{ m}\) from the ground level. So, the total vertical distance moved by water is \(20 + 10 = 30 \text{ m}\). The force required to lift the water is the weight of the water, and the work done is force times distance. Let's first convert the volume of the water to mass. We know 1 liter of water has a mass of 1 kg, so the mass of water is \(22380 \text{ kg}\). The work done \(W\) is calculated as follows:\[W = ext{mass} \times g \times ext{height} = 22380 \text{ kg} \times 10 \text{ m/s}^2 \times 30 \text{ m} = 6714000 \text{ J}\].

Step 2: Convert Motor Power to Watts

The power of the motor is given in horsepower (HP). First, convert this to watts. We know that \(1 \text{ HP} = 746 \text{ W}\), so:\[10 \text{ HP} = 10 \times 746 \text{ W} = 7460 \text{ W}\].

Step 3: Calculate Time Using Power and Work Done

The relation between power, work done, and time is given by the formula:\[P = \frac{W}{t}\], where:- \(P\) is the power in watts, - \(W\) is the work done in joules, - \(t\) is the time in seconds.Rearranging the formula to solve for time, we get:\[t = \frac{W}{P} = \frac{6714000 \text{ J}}{7460 \text{ W}} \approx 900 \text{ seconds}\].

Step 4: Convert Time to Minutes

Convert the calculated time from seconds to minutes by dividing by 60, since there are 60 seconds in a minute.\[t = \frac{900}{60} = 15 \text{ minutes} \].

Key Concepts

Power conversion

When dealing with machinery, many motors and engines have their power rated in horsepower (HP), especially in fields involving older technology or information sources. However, for scientific and engineering calculations, it's often more practical to use watts.

• 1 horsepower is equivalent to 746 watts.
• To convert horsepower to watts, you simply multiply the number of horsepower by 746.
• This is handy when calculating energy or power usage over time, as watts are a more standard unit in these equations.

Understanding this conversion is crucial when determining how much energy a motor uses or how much work it can perform over time. By converting motor power from HP to watts, we can accurately compute the number of joules of energy used, linking back to the basic definition of power as energy per unit time.

Vertical motion

Vertical motion is crucial to understand when discussing work done in lifting. It refers to the movement of an object in a straight line along the vertical axis, like when water is being pumped upwards.

• This movement is against gravity, meaning an external force must do work to move the object upward.
• The work done is calculated as the product of the force needed to move the object (weight in this case) and the distance moved against gravity.
• In our example, we moved water from 20 meters to a height of 10 meters, totaling 30 meters of vertical motion.

Understanding vertical motion helps us calculate the energy needed to perform tasks against gravitational forces.

Gravitational potential energy

Gravitational potential energy (GPE) is the energy stored in an object due to its height above the ground. When lifting an object, like water being pumped, this energy increases as the object is raised.

• GPE is calculated by the formula: \[ GPE = ext{mass} imes g imes ext{height} \]
• In our scenario, the mass of the water is 22,380 kg, and it is raised to a height of 30 meters. With gravity constant (\( g = 10 ext{ m/s}^2 \) ), we find the GPE involved.
• This represents the total work done and corresponds to the energy required to lift the mass to the specified height.

Gravitational potential energy is essential in determining the effort needed to elevate objects and provides a clear understanding of energy transfers in systems involving height changes.