The crate will not slip as long as it has the same acceleration as the truck. Therefore, a net force must act on the crate to accelerate it, and the static frictional force
![](../../cutnell9780470879528/c04/math/math257.gif)
contributes to this net force. Since the crate tends to slip backward, the static frictional force is directed forward, up the hill. As the acceleration of the truck increases,
![](../../cutnell9780470879528/c04/math/math257.gif)
must also increase to produce a corresponding increase in the acceleration of the crate. However, the static frictional force can increase only until its maximum value
![](../../cutnell9780470879528/c04/math/math258.gif)
is reached, at which point the crate and truck have the maximum acceleration
![](../../cutnell9780470879528/c04/math/math259.gif)
. If the acceleration increases even more, the crate will slip. To find
![](../../cutnell9780470879528/c04/math/math259.gif)
, we will employ Newton's second law, the definition of weight, and the relationship between the maximum static frictional force and the normal force.