Question

An automobile is moving at $100km{ h }^{ -1 }$ and is exerting an attractive force of 400 kgf. What horsepower must the engine develop if $20\%$ of the power developed is wasted?

Answer 1

To solve this problem, first find the output power of the engine. This can be calculated by taking the product of force and velocity. Then, find the efficiency of the engine by subtracting the power wasted from $100\%$. Use the formula for efficiency, substitute the values of efficiency and output power in it. This will give the input power. But this power will be in Watt. So, convert it in horse power using the conversion factor. This obtained power the engine must develop.

Complete answer:
Given: Speed of an automobile (v)= $100km{ h }^{ -1 }$
Attractive force (F)= 400 kgf
1 kgf is equal to 9.8 N. Therefore, 400 kgf is 3920 N. So, the attractive force is 3920 N.
$\Rightarrow F= 3920 N$
$v=100km{ h }^{ -1 }= 100\times \dfrac { 1000 }{ 60\times 60 }= \dfrac { 250 }{ 9 } { m }/{ s }$
Now, it is given that $20\%$ of the power developed is wasted. Therefore, $80\%$ is used.
$\therefore \eta= \dfrac {80}{100}$
Output power of the engine is given by,
${ P }_{ out }=\quad F\times v$
Substituting the values in above equation we get,
$\Rightarrow { P }_{ out }=\quad 3920 \times \dfrac { 250 }{ 9 }$
$\Rightarrow {P}_{out}= 10.9 \times {10}^{4} Watt$
We know, efficiency is given by,
$\eta =\cfrac { { P }_{ out } }{ { P }_{ in } }$
Substituting the values in above equation we get,
$\dfrac { 80 }{ 100 } =\dfrac { 10.9 \times {10}{4} }{ { P }_{ in } }$
$\Rightarrow 0.80=\dfrac { 10.9 \times {10}^{4} }{ { P }_{ in } }$
$\Rightarrow {P }_{ in }=\dfrac { 10.9 \times {10}^{4} }{ 0.80 }$
$\Rightarrow {P}_{in}= 13.6 \times {10}^{4} Watt$
We know, 1 h.p.= 746 Watt. Therefore, $13.6 \times {10}^{4}$ Watt is,
${ P }_{ in }=\dfrac { 13.6\times { 10 }^{ 4 } }{ 746 }$
$\Rightarrow {P}_{in} = 182.45 h.p$

Note:
Students should take care of the conversion of units. Velocity is given in terms of kmph so, convert it in m/s. While the force is in kgf convert it into newton (N). After calculating the input power make sure you convert it into horsepower as it is already mentioned in the question. If it was not mentioned then you could have left it in terms of Watt (W).