Question

An engine can pump $30000L$ water to a vertical height of $45m$ in $10\min$. Calculate the work done by the machine and its power. (Given the density of water ${10^3}kg.{m^{ - 3}}$)

Answer 1

First, we need to understand how to determine the work done by a machine and the relation between a machine's power and work done by a machine. The displacement done by a force measures the work done in a physical system. The power of a machine is given as the work done in a unit of time.

Complete step by step solution:
In physical science, work is referred to as the energy transferred from one object to another by applying a force. Work is represented as the product of applied force and displacement in the direction of that force. Work is a scalar quantity, although it is a product of two vector quantities.
In the simplest form, it is written as-
$W = F{{. }}s$
where,
$F$ is the applied force
$s$ is the displacement along the direction of the applied force
Given,
The volume of water,
$V = 30000L = 30000 \times {10^{ - 3}}{m^3}$ $\left[ {\because 1L = {{10}^{ - 3}}{m^3}} \right]$
The density of water, $\rho = {10^3}kg.{m^{ - 3}}$
Therefore, the mass of the given water-
$M = \rho \times V$
$\Rightarrow M = 30000 \times {10^{ - 3}} \times {10^3}kg$
$\Rightarrow M = 30000kg$
We know work done by a machine is given by-
$W = M.g.h$ …………….$(1)$
where,
$g$ is the gravitational acceleration
$h$ is the displacement done by the force
Therefore, we put $g = 9.8m{s^{ - 2}}$ and $h = 45m$ in the equation $(1)$-
$\Rightarrow W = \left( {30000 \times 9.8 \times 45} \right)J$
$\Rightarrow W = 13.23 \times {10^6}J$
Therefore, the total work done by the machine is $13.23 \times {10^6}J$.
Given, the machine can pump the given amount of water in $10\min$.
Therefore, the time,
$\Rightarrow t = 10 \times 60\sec$
$\Rightarrow t = 600\sec$
Now the power of a machine is given by-
$P = \dfrac{W}{t}$
We put $W = 13.23 \times {10^6}J$ and $t = 600\sec$ in the above equation and get-
$\Rightarrow P = \dfrac{{13.23 \times {{10}^6}}}{{600}}W$
$\Rightarrow P = 22050W$

Therefore, the power of the given machine is $22050W$.

Note: The most vital quantity related to a machine in a physical system is its mechanical efficiency. The efficiency of a machine can be given by the ratio of power output and power input. Mechanical efficiency is a dimensionless quantity that measures the effectiveness of the machine.