Suppose that you are standing on a train accelerating at 0.39g. What minimum coefficient of static friction must exist between your feet and the floor if you are not to slide?
µ = v^2 / Rg
To determine the minimum coefficient of static friction required to prevent sliding on a train accelerating at 0.39g, we need to consider the forces acting on you.
First, let's identify the forces involved:
1. The force of gravity acting vertically downwards, which can be calculated as: F_gravity = mass * gravity.
2. The normal force exerted by the floor, which is equal in magnitude and opposite in direction to the force of gravity when no sliding occurs.
3. The force of static friction between your feet and the floor, which opposes the tendency for you to slide.
Next, let's analyze the situation using Newton's second law:
The net force acting on you in the horizontal direction is given by: F_net = mass * acceleration.
The force of static friction can also be represented as: F_friction = coefficient * normal force.
Since the normal force is equal to the force of gravity on a level surface, we can rewrite the equation for the force of static friction as: F_friction = coefficient * F_gravity.
Now, substituting the given values:
F_net = mass * acceleration
F_friction = coefficient * F_gravity
Given that the acceleration on the train is 0.39g, where g is the acceleration due to gravity (9.8 m/s^2), we can rewrite the equation for F_net as: F_net = mass * 0.39 * g.
To avoid sliding, the maximum static friction force must equal the net force acting on you, which is given by F_net. Therefore, we can equate the two equations:
mass * 0.39 * g = coefficient * mass * g
The masses cancel out, leaving:
0.39 = coefficient
Therefore, the minimum coefficient of static friction required to prevent sliding is 0.39.