Final answer:
To prevent a crate from slipping on the bed of a truck, the truck's acceleration must not exceed the maximum static frictional force divided by the mass of the crate. Given the coefficient of static friction and the crate's mass, you can calculate the maximum static frictional force and thereby determine the maximum acceleration that keeps the crate from slipping.
Step-by-step explanation:
The question is concerned with calculating the maximum acceleration a truck can achieve without causing a crate to slip backward on its surface. When the truck accelerates, it exerts a force on the crate through static friction. This force is determined by the coefficient of static friction (μs) and the normal force (N) acting on the crate, given by μs N. To ensure the crate does not slip, the maximum force of static friction must be greater than or equal to the force required to accelerate the crate at the same rate as the truck.
Let's consider a scenario where the coefficient of static friction between the crate and the truck bed is 0.400 and the crate has a mass of 50.0 kg. The normal force (N) acting on the crate is equal to its weight, which can be calculated as the mass (m) times the acceleration due to gravity (g), N = m × g. With the given values, N = (50.0 kg)(9.8 m/s²) = 490 N. Thus, the maximum static frictional force (fs max) that can prevent slipping is fs max = μs × N = (0.400)(490 N) = 196 N.
If the truck accelerates at 5.00 m/s², the force required to accelerate the crate at this rate would be F = ma = (50.0 kg)(5.00 m/s²) = 250 N. Since 250 N is greater than 196 N, the static friction limit, the crate will slip. If we want to prevent the crate from slipping, we must limit the truck's acceleration to a value that results in a required force less than or equal to 196 N.