Question 72: Work, Energy and Power & Circular Motion

Question

The ratio of powers of two motors is $$\frac{3 \sqrt{x}}{\sqrt{x}+1}$$, that are capable of raising $$300 \mathrm{~kg}$$ water in 5 minutes and $$50 \mathrm{~kg}$$ water in 2 minutes respectively from a well of $$100 \mathrm{~m}$$ deep. The value of $$x$$ will be

Accepted Answer

Let us first find the power required to lift the water using each motor. Let $P_1$ be the power of the first motor, and $P_2$ be the power of the second motor.
The work done in lifting the water is given by $W = mgh$, where $m$ is the mass of water lifted, $g$ is the acceleration due to gravity, and $h$ is the height through which the water is lifted. In this case, $m = 300\mathrm{~kg}$ and $h = 100\mathrm{~m}$ for the first motor, and $m = 50\mathrm{~kg}$ and $h = 100\mathrm{~m}$ for the second motor.
The work done in lifting the water in 5 minutes by the first motor is:
$$W_1 = mgh = (300\mathrm{~kg})(9.8\mathrm{~m/s^2})(100\mathrm{~m}) = 294000\mathrm{~J}$$
The power required to do this work in 5 minutes is:
$$P_1 = \frac{W_1}{t_1} = \frac{294000\mathrm{~J}}{300\mathrm{~s}} = 980\mathrm{~W}$$
The work done in lifting the water in 2 minutes by the second motor is:
$$W_2 = mgh = (50\mathrm{~kg})(9.8\mathrm{~m/s^2})(100\mathrm{~m}) = 49000\mathrm{~J}$$
The power required to do this work in 2 minutes is:
$$P_2 = \frac{W_2}{t_2} = \frac{49000\mathrm{~J}}{120\mathrm{~s}} = 408.33\mathrm{~W}$$
The ratio of the powers of the two motors is:
$$\frac{P_1}{P_2} = \frac{980\mathrm{~W}}{408.33\mathrm{~W}} \approx 2.4$$
We are given that this ratio is equal to:
$$\frac{3 \sqrt{x}}{\sqrt{x}+1}$$
We can solve for $x$ as follows:
$$\frac{3 \sqrt{x}}{\sqrt{x}+1} = 2.4$$
$$3\sqrt{x} = 2.4(\sqrt{x}+1)$$
$$3\sqrt{x} = 2.4\sqrt{x} + 2.4$$
$$(3-2.4)\sqrt{x} = 2.4$$
$$0.6\sqrt{x} = 2.4$$
$$\sqrt{x} = 4$$
$$x = 16$$
Therefore, the value of $x$ is 16.

Answer

16

Answer

4

Answer

2

Answer

2.4

Work, Energy and Power & Circular Motion

The ratio of powers of two motors is $\frac{3 \sqrt{x}}{\sqrt{x}+1}, that are capable of raising 300 \mathrm{~kg} water in 5 minutes and 50 \mathrm{~kg} water in 2 minutes respectively from a well of 100 \mathrm{~m} deep. The value of x$ will be

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