Magnitude of the tension force

Classical Mechanics Level 3

A boy is pulling a sled loaded with mass $m = 20\text{ kg}$ on some horizontal, snowy pavement. The velocity of the sled is constant. The coefficient of kinetic friction $μ_k$ between the runner and the pavement is $0.1$ and the angle $φ$ between the cord and the pavement is $30^\circ.$ Find the magnitude of the tension force by which the boy affects the sled.

Details and assumptions:

Take $g = 9.81 ~ \text{m/s}^2.$
Give your answer in Newton to two decimal places.

The answer is 21.41.

1 solution

Muhammad Erfan
Oct 15, 2017

the force diagram:

The velocity of the sledge is constant so the acceleration of the sledge equal to zero in the direction of x axis or $\sum { F_{x} } =0$

$\sum { { F }_{ x } } =T\cos { 30° } -f=0$

Normal Force acting on the sledge: $N=mg-T\sin { 30° }$

$f={ \mu }_{ k }N={ \mu }_{ k }\left( mg-T\sin { 30° } \right) =T\cos { 30° }$

$f=0.1\left[ \left( 20\times 9.81 \right) -T\sin { 30° } \right] =T\cos { 30° }$

$f=19.62-T\left( 0.1\sin { 30° } \right) =T\cos { 30° }$

$T=\frac { 19.62 }{ \left( 0.1\sin { 30°+\cos { 30° } } \right) } \approx \boxed{21.41}$

so, the tension force by which the boy affects the sledge is $21.42$ newton