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

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To determine the maximum acceleration of a truck without a box sliding forward, the key is understanding the forces at play. The static friction force, which prevents the box from sliding, is equal to the product of the coefficient of static friction and the normal force. When the truck decelerates, this friction must counteract the forward motion of the box, leading to the equation f_s = μ*n = μ*mg. The confusion arises from the relationship between acceleration and the forces acting on the box, particularly in how to express the maximum deceleration without exceeding the static friction limit. Establishing the correct equation will clarify how to solve for the truck's maximum acceleration while ensuring the box remains stationary.
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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?
 
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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?
 
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