Maximum Acceleration of Train to Keep Box Stationary

Introduction:
To determine the maximum acceleration of a train in which a box lying on its floor will remain stationary, we need to consider the coefficient of static friction between the box and the train's floor. The coefficient of static friction, denoted by 's', measures the maximum frictional force that can exist between two surfaces without causing relative motion.

Formula:
The maximum static frictional force (F) between the box and the train's floor can be calculated using the formula:
F = μ * N
where μ is the coefficient of static friction and N is the normal force acting on the box.

Normal Force:
The normal force (N) is the force exerted by a surface to support the weight of an object resting on it. In this case, the weight of the box is balanced by the normal force, which is equal in magnitude but opposite in direction. Therefore, the normal force can be calculated as:
N = m * g
where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Maximum Acceleration:
To determine the maximum acceleration of the train, we need to equate the maximum static frictional force to the product of mass and acceleration (F = m * a). Rearranging the equation, we have:
μ * N = m * a

Substituting the values of N and rearranging, we get:
μ * m * g = m * a

Simplifying the equation, we find:
a = μ * g

Calculation:
Given that the coefficient of static friction (μ) between the box and the train's floor is 0.15, and the acceleration due to gravity (g) is approximately 9.8 m/s^2, we can calculate the maximum acceleration as follows:
a = 0.15 * 9.8
a ≈ 1.47 m/s^2

Conclusion:
Therefore, the maximum acceleration of the train in which the box lying on its floor will remain stationary is approximately 1.47 m/s^2. If the train accelerates beyond this value, the static frictional force between the box and the train's floor will be exceeded, causing the box to slide.