6.2 Friction  (Page 7/12)

Snowboarding

Earlier, we analyzed the situation of a downhill skier moving at constant velocity to determine the coefficient of kinetic friction. Now let’s do a similar analysis to determine acceleration. The snowboarder of [link] glides down a slope that is inclined at $\theta ={13}^0$ to the horizontal. The coefficient of kinetic friction between the board and the snow is ${\mu }_{\text{k}}=0.20.$ What is the acceleration of the snowboarder?

Strategy

The forces acting on the snowboarder are her weight and the contact force of the slope, which has a component normal to the incline and a component along the incline (force of kinetic friction). Because she moves along the slope, the most convenient reference frame for analyzing her motion is one with the x -axis along and the y -axis perpendicular to the incline. In this frame, both the normal and the frictional forces lie along coordinate axes, the components of the weight are $mg\text{sin}\theta \text{along the slope and}mg\text{cos}\theta \text{at right angles into the slope}$ , and the only acceleration is along the x -axis $\left(a_y=0\right).$

Solution

We can now apply Newton’s second law to the snowboarder:

From the second equation, $N=mg\text{cos}\theta .$ Upon substituting this into the first equation, we find

Significance

Notice from this equation that if $\theta$ is small enough or ${\mu }_{\text{k}}$ is large enough, $a_x$ is negative, that is, the snowboarder slows down.

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