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.

(b) The truck barely exceeds this acceleration and then moves with constant acceleration, with the crate sliding along its bed. What is the acceleration of the crate relative to the ground?

(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 acceleration of the crate relative to the ground, we need to first understand the forces acting on the crate.

1. Calculate the maximum static friction force (Fstatic_max) between the crate and the flatbed:
Fstatic_max = coefficient of static friction (μstatic) * normal force (N)

The normal force (N) is equal to the weight of the crate, which is given by:
N = m * g, where m is the mass of the crate and g is the acceleration due to gravity (approximately 9.8 m/s^2).

In this case, m = 55.6 kg, so:
N = 55.6 kg * 9.8 m/s^2

Now, we can calculate the maximum static friction force:
Fstatic_max = 0.330 * (55.6 kg * 9.8 m/s^2)

2. Determine if the maximum static friction force is sufficient to provide the required acceleration. If not, the crate will start sliding and kinetic friction will come into play.

Calculate the maximum static friction force required to accelerate the crate:
Fstatic_required = m * a, where a is the desired acceleration.

In this case, m = 55.6 kg and Fstatic_required = Fstatic_max.
Therefore, Fstatic_max = 55.6 kg * a.

If Fstatic_max is greater than or equal to the maximum static friction force calculated earlier, the crate will not slide and the acceleration will be the same as the acceleration of the truck.

3. If the maximum static friction force is not sufficient, the crate will start sliding, and the kinetic friction force will come into play. In this case, we need to calculate the kinetic friction force (Fkinetic) and the acceleration (akinetic).

The kinetic friction force is given by:
Fkinetic = coefficient of kinetic friction (μkinetic) * N

In this case, μkinetic = 0.320 and N is still equal to 55.6 kg * 9.8 m/s^2.
Therefore, Fkinetic = 0.320 * (55.6 kg * 9.8 m/s^2)

The acceleration of the crate relative to the ground (akinetic) is:
akinetic = (Fnet - Fkinetic) / m

Here, the net force (Fnet) on the crate is equal to the force applied by the truck minus the kinetic friction force. If the truck provides a constant acceleration (a_constant), the net force is equal to:
Fnet = m * a_constant

Substitute the values into the equation:
akinetic = (m * a_constant - Fkinetic) / m

Simplifying the equation gives:
akinetic = a_constant - (Fkinetic / m)

Therefore, if the maximum static friction force (Fstatic_max) is sufficient to provide the required acceleration, the acceleration of the crate relative to the ground is equal to the acceleration of the truck (a_constant).

If the maximum static friction force is not sufficient, the acceleration of the crate relative to the ground will be the constant acceleration of the truck (a_constant) minus the kinetic friction force (Fkinetic) divided by the mass of the crate (m).