Question

A 2400 W engine pulls a 200 kg block at constant speed up a 12.0 m long, 25.0° incline.
Determine long does it takes to cover this distance.

Answer 1

Answer on Question #40095 - Physics - Mechanics

1. A 2400 W engine pulls a 200 kg block at constant speed up a 12.0 m long, 25.0° incline. Determine long does it takes to cover this distance.

$$\begin{array}{l} P = 2400\,W \\ m = 200\,kg \\ l = 12\,m \\ \alpha = 25{}^{\circ} \\ t - ? \end{array}$$

Solution.

Let write down the equation of the motion of the body (Newton second law):

$$\dot{m}a = m\dot{g} + \dot{N} + \dot{F},$$

where $\dot{F}$ is the pulling force.

As the velocity of the block is constant, the acceleration equals to zero.

Then, write down this law in projectives onto the $X$- and $Y$-axes:

$$\begin{array}{l} X: [m \cdot 0 = mg \sin \alpha - F \\ Y: [m \cdot 0 = -mg \cos \alpha + N] \end{array}$$

so the pulling force is $F = mg \sin \alpha$.

At the constant speed, the power of the engine is $P = F \cdot v$.

We can find the time of the motion of the block with the constant velocity:

$$t = \frac{l}{v} = l: \frac{P}{F}, \quad t = \frac{lm g \sin \alpha}{P}.$$

Let check the dimension.

$$[t] = \frac{m \cdot kg \cdot \frac{m}{s^2}}{W} = \frac{m \cdot N}{J / s} = s.$$

Let evaluate the quantity.

$$t = \frac{12 \cdot 200 \cdot 9.8 \cdot \sin 25{}^{\circ}}{2400} = 4.14(s).$$

Answer: 4.14 s.