Question

A farm tractor tows a 3850 kg trailer up a 13 degrees incline with a steady speed of 3.0 m/s. What force does the tractor exert on the trailer? (Ignore friction.)

Answer 1

Concepts and reason

The concepts required to answer the given question is the force.

The force is a vector quantity. Force is defined as the rate of change of momentum. Since, momentum is vector quantity. So, force is also a vector quantity.

The vector quantities or vectors are those physical quantities which have magnitude as well as direction. The law of basic algebra is not applicable for the vectors. The vectors follow the law of vectors for addition, subtraction and multiplication.

Fundamentals

A general form of a vector in the form of its components in 2-dimension is,

$\vec A = {A_x}\hat i + {A_y}\hat j$

Here, ${A_x}$is component of vector along x-axis and ${A_y}$ is vector along y-axis.

The force can also be written in component form as,

$\vec F = {F_x}\hat i + {F_y}\hat j$

Here, ${F_x}$ is the x- component of force, ${F_y}$ is y-component of the force.

The magnitude of force acting on a body under acceleration due to gravity is,

$F = mg$

If force act on a body with making an angle $\theta$ with positive x axis in counterclockwise direction. Then the components of force are given by,

$\begin{array}{c}\\{F_x} = F\cos \theta \\\\{F_y} = F\sin \theta \\\end{array}$

Since, the tractor moves by steady velocity. So, there is no acceleration. The force acts on the trailer is due to the weight of the tractor. It is y-component of the force.

So, force on exert on the trailer is,

${F_y} = F\sin \theta$

Substitute mg for F in above equation as follows:

${F_y} = mg\sin \theta$

Substitute $3850 {\rm{kg}}$ for $m$ , $9.8 {\rm{m/}}{{\rm{s}}^2}$ for $g$and $13^\circ$ for $\theta$ in above equation as follows:

$\begin{array}{c}\\F = \left( {3850 {\rm{kg}}} \right)\left( {9.8 {\rm{m/}}{{\rm{s}}^2}} \right)\sin \left( {13^\circ } \right)\\\\ = 8487.4 {\rm{N}}\\\end{array}$

Ans:

The force that tractor exert on the trailer is $8487.4 {\rm{N}}$.