Part A
A 15 kg box is sitting in the bed of a pickup truck. The truck begins to accelerate at a constant rate of 3.5 m/s 2. Given that the friction between the box and the truck bed is characterized by a coefficient of kinetic friction of 0.25 and a coefficient of static friction equal to 0.40, what is the magnitude of the friction force acting on the box once the truck begins its acceleration?
System: We will treat the box as a point particle subject to external influence from the earth (gravity) and the truck bed (normal force and friction force).
Model: Point Particle Dynamics.
Approach: We begin with a sketch that represents the situation, and then create the appropriate free body diagram.
It is important to note that friction works to prevent movement of the interface between the box and the truck bed. The truck bed is moving forward, so friction will attempt to pull the box forward as well. If the box moves at the same rate as bed, then the interface is static. For this reason, "static" friction will actually cause motion of the box in this case!
Using the free-body diagram, we construct the equations of Newton's Second Law applied to the box:
\begin
[\sum F_
= F_
= ma_
][\sum F_
= N - mg = ma_
] \end
Since the truck is moving only in the x-direction, we expect ay = 0. Thus, we know that the normal force acting on the box will equal it weight. If friction is adequate, we expect that the box will accelerate in the x-direction at the same rate as the truck does. In that case, we expect:
\begin
[ F_
= (15\:
)(3.5\:
^
) = 53\:
]\end
We are not finished yet. It is important now to check that this result does not conflict with the requirement that
\begin
[ F_
\le \mu_
N] \end
Since we have already used the y-direction equation of Newton's Second Law to conclude that the normal force on the box is equal to mg, we find:
\begin
[ F_
= 53\:
< \mu_
N = 0.40(15\:
)(9.8\:
^
) = 59\:
]\end
We therefore conclude that our answer, Ff = 53 N, is compatible with the static friction limit.
Part B
Consider the same basic situation as above, but now suppose that the truck accelerates at a rate of 4.0 m/s 2 rather than 3.5 m/s 2. If the center of mass of the box is located 2.0 m horizontally from the edge of the truck bed, how much time will elapse from the instant the truck begins to accelerate until the instant the box falls off the truck bed?
Approach: We use the same system and model as in Part A. Once again, friction is trying to keep the box moving at the same rate as the truck bed.