Question

A man cycles up a hill rising 1 meter vertically for every 50 meters along the slope. Find the power of the man, if he cycles up at the rate of $3 \cdot 6\dfrac{{km}}{h}$. The weight of the cycle and man is equal to 120kg. Neglect force of friction:
A) 32.25 watt.
B) 24 watt.
C) 25.32 watt.
D) 52.32 watt.

Answer 1

The climbing up on the hill can be considered as climbing on an inclined plane. The acceleration on the inclined plane is different from the normal acceleration. The power is defined as the rate of change of work done with respect to time.

Formula used: The formula of the power is given by,
$\Rightarrow P = F \times v$
Where power is equal to P, the force is equal to F and the velocity of the body is equal to v.
The acceleration at the inclined plane is given by,
$\Rightarrow a = g\sin \theta$
Where acceleration is the angle of the inclined hill is $\theta$ and the acceleration due to gravity is g.

Complete step by step answer:
It is given in the problem that a man cycles up a hill rising 1 meter vertically for every 50 meters along the slope. If the cycles up at the rate of $3 \cdot 6\dfrac{{km}}{h}$, the weight of the cycle and man is equal to 120kg and we need to find the power of the man.
First of all let us convert the speed of man from $\dfrac{{km}}{h}$ to $\dfrac{m}{s}$.
Speed of man is equal to,
$\Rightarrow {v_m} = 3 \cdot 6 \times \dfrac{5}{{18}}$
$\Rightarrow {v_m} = 3 \cdot 6 \times \dfrac{5}{{18}} = 1\dfrac{m}{s}$
The speed of the man is equal to ${v_m} = 1\dfrac{m}{s}$.
As it is given that for every 50 meters on the slope there is 1 meter rising vertically up which means that the sine of angle is equal to,
$\Rightarrow \sin \theta = \dfrac{1}{{50}}$
The acceleration at the inclined plane is given by,
$\Rightarrow a = g\sin \theta$
Where acceleration is a, the angle of the inclined hill is $\theta$ and the acceleration due to gravity is g.
The acceleration felt by the man is equal to,
$\Rightarrow a = g\sin \theta$
The acceleration due to gravity is equal to $g = 10\dfrac{m}{{{s^2}}}$ and $\sin \theta = \dfrac{1}{{50}}$ therefore we get,
$\Rightarrow a = g\sin \theta$
$\Rightarrow a = 10 \times \dfrac{1}{{50}}$
$\Rightarrow a = 0 \cdot 2\dfrac{m}{{{s^2}}}$
The force on the man is equal to,
$\Rightarrow F = m \times a$
The weight of the man and cycle is equal to120kg and acceleration is equal to $a = 0 \cdot 2\dfrac{m}{{{s^2}}}$.
$\Rightarrow F = 120 \times 0 \cdot 2$
$\Rightarrow F = 24N$
The force on the man is equal to $F = 24N$.
The formula of the power is given by,
$\Rightarrow P = F \times v$
Where power is equal to P, the force is equal to F and the velocity of the body is equal to v.
The force on the man is equal to $F = 24N$ and the velocity of the man is equal to ${v_m} = 1\dfrac{m}{s}$.
The power of the man is equal to,
$\Rightarrow P = F \times v$
$\Rightarrow P = 24 \times 1$
$\Rightarrow P = 24{\text{watt}}$.
The power of the man is equal to $P = 24{\text{watt}}$.

The correct answer for this problem is option B.

Note: The students are advised to understand and remember the formula of the power and also in different terms as it is very helpful in the solving problem like these. In this problem we use the formula of the power in terms of force and velocity.