Question

A person whose mass is m = 70.0 kg steps on a mechanical bathroom scale placed on an inclined plane that makes the angle α = 12.9° with the horizontal. What is the reading on the scale?

Answer 1

A person whose mass is $m = 70.0 \, \text{kg}$ steps on a mechanical bathroom scale placed on an inclined plane that makes the angle $\alpha = 12.9{}^\circ$ with the horizontal. What is the reading on the scale?

**Solution:**

$m = 70.0 \, \text{kg}$ - mass of the person;

$\alpha = 12.9{}^{\circ}$ - angle which plane makes with the horizontal;

N - reaction force;

M - mass of the person which shows the scales.

Newton's second law for person (the first law of equilibrium):

$$\overrightarrow{\mathrm{mg}} + \overrightarrow{\mathrm{N}} + \overrightarrow{\mathrm{P}}_{\text{friction}} = \overrightarrow{0}$$ $$y: N - m g_y = 0$$

From the a right triangle:

$$\cos \alpha = \frac{m g}{m g_y}$$ $$m g_y = m g \cdot \cos \alpha$$

(2)in(1):

$$N = m g_y = m g \cdot \cos \alpha$$

Newton's third law: When person exerts a force on a plane, the plane simultaneously exerts a force equal in magnitude and opposite in direction to that of the person:

$$\overrightarrow{\mathrm{N}} = - \overrightarrow{\mathrm{P}}$$ $$N = P$$ $$P = M g$$

(5)and(3)in(4):

$$m g \cdot \cos \alpha = M g$$ $$M = m \cdot \cos \alpha = 70 \, \text{kg} \cdot \cos 12.9{}^{\circ} = 68.23 \, \text{kg}$$

Answer: the reading on the scale is $68.23 \, \text{kg}$