A crate of mass 55.6 kg is being transported on the flatbed of a pickup truck. The coefficient of static friction between the crate and the trucks flatbed is 0.330, and the coefficient of kinetic friction is 0.320.


(a) The truck accelerates forward on level ground. What is the maximum acceleration the truck can have so that the crate does not slide relative to the trucks flatbed?

(a)

ma=F(fr) = μ(s) •N= μ(s) •m•g,
a= μ(s)•g,
(b)
ma1=F1(fr) = μ(k) •N= μ(k) •m•g,
a1= μ(k)•g,

To determine the maximum acceleration that the truck can have without causing the crate to slide relative to the truck's flatbed, we need to consider the force of friction acting on the crate.

First, let's calculate the maximum static friction force that can be exerted on the crate.

The formula for static friction force is given by:

\[f_{\text{static}} = \mu_s \cdot N\]

where \(\mu_s\) is the coefficient of static friction and \(N\) is the normal force between the crate and the truck's flatbed.

The normal force (\(N\)) is equal to the weight of the crate (\(mg\)), where \(m\) is the mass of the crate and \(g\) is the acceleration due to gravity (\(9.8 \, \text{m/s}^2\)).

So, the maximum static friction force is:

\[f_{\text{static}} = \mu_s \cdot m \cdot g\]

Substituting the given values:

\[f_{\text{static}} = 0.330 \cdot 55.6 \, \text{kg} \cdot 9.8 \, \text{m/s}^2\]

Now, the maximum static friction force acting on the crate can be used to find the maximum acceleration.

The force of friction (\(f_{\text{static}}\)) is given by:

\[f_{\text{static}} = m \cdot a\]

where \(a\) is the maximum acceleration.

Rearranging the equation to solve for \(a\):

\[a = \frac{f_{\text{static}}}{m}\]

Substituting the known values:

\[a = \frac{0.330 \cdot 55.6 \, \text{kg} \cdot 9.8 \, \text{m/s}^2}{55.6 \, \text{kg}}\]

After evaluating the expression, the maximum acceleration is approximately 3.26 m/s^2.

Therefore, the maximum acceleration the truck can have so that the crate does not slide relative to the truck's flatbed is approximately 3.26 m/s^2.