What is the maximum acceleration a truck can have without a box sliding forward?

[Metalsonic75]

1. A person driving a truck has a box in the back. They are driving at a constant speed (v_0) when they notice a red light and have to stop. What is the maximum acceleration the truck can have without causing the box to slide forward? (Answer must be given in terms of variables, as no values are given)



2. Sum of forces in the x direction=m*a, f_s=$$\mu$$*n= $$\mu$$*mg.



3. The thing that confuses me most is the acceleration. I assume the acceleration is negative because the truck is slowing down. I don't understand what force is acting on the box to slide it forward (I know friction is pushing it back), or how to measure that force. I assume f_s has to equal that force so the box doesn't move, but if it does, the sum of forces = 0, and thus m*a=0, thus a=0. But a can't be zero! So how do I solve for a?

 

[Kurdt]

The net force is zero and the net force is not just ma. You stated that you knew that the static friction force must be equal to the maximum stopping force. Can you set up an equation that will allow you to solve for acceleration now?

 

[ogb p]

I would approach this problem by first identifying all of the forces acting on the box. These forces include the force of gravity pulling the box downward (mg), the normal force from the truck bed pushing the box upward (n), and the force of friction (f_s) between the box and the truck bed.

Next, I would use the equation of motion in the x-direction (sum of forces = mass times acceleration) to determine the maximum acceleration that the truck can have without causing the box to slide forward. This equation can be written as:

m*a = f_s - \mu*n

Since the truck is driving at a constant speed (v_0), we can assume that the acceleration (a) is equal to zero. This means that the sum of forces in the x-direction must also equal zero.

Therefore, we can set the equation equal to zero and solve for the maximum acceleration (a) as follows:

0 = f_s - \mu*n

a = \frac{f_s}{m} = \frac{\mu*n}{m}

We know that the mass of the box (m) and the coefficient of friction (\mu) are constants, so the only variable left to solve for is the normal force (n). This normal force is dependent on the weight of the box (mg) and the angle of the truck bed.

To determine the maximum acceleration, we would need to know the specific values for the weight of the box and the angle of the truck bed. Without these values, we can only provide a general equation for the maximum acceleration, as shown above.

In conclusion, to prevent the box from sliding forward, the maximum acceleration of the truck can be calculated using the equation a = \frac{\mu*n}{m}, where n is the normal force and is dependent on the weight of the box and the angle of the truck bed.