Suppose that you are standing on a train accelerating at 0.33g. What minimum coefficient of static friction must exist between your feet and the floor if you are not to slide?
To determine the minimum coefficient of static friction needed to prevent sliding on a train accelerating at 0.33g, we can use Newton's second law of motion. This law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, let's consider the person standing on the train as the object. The net force acting on the person can be calculated by multiplying their mass (m) by the acceleration due to gravity (g) plus the acceleration of the train (0.33g).
Net force = (m) * (g + 0.33g)
Now, the force of static friction can be represented as the coefficient of static friction (μs) multiplied by the normal force (N) exerted on the person.
Force of static friction = μs * N
The normal force (N) is equal to the weight of the person, which can be calculated by multiplying their mass (m) by the acceleration due to gravity (g).
N = m * g
Since the person on the train is not sliding, the force of static friction must be equal to the net force.
Therefore, μs * N = (m) * (g + 0.33g)
Now, we can substitute the value of N in terms of m and g:
μs * (m * g) = (m) * (g + 0.33g)
Simplifying the equation by canceling out the mass:
μs * g = g + 0.33g
Combining like terms:
μs = 1 + 0.33
μs = 1.33
Therefore, the minimum coefficient of static friction that must exist between your feet and the floor to prevent sliding on a train accelerating at 0.33g is 1.33.
To determine the minimum coefficient of static friction required to prevent sliding, we can use the equation:
Frictional force = coefficient of static friction x Normal force
In this case, the normal force acting on you is equal to your weight:
Normal force = mass x acceleration due to gravity
To find the normal force, we need to know your mass. Let's assume your mass is 70 kilograms.
1. Calculate the normal force:
Normal force = mass x acceleration due to gravity
Normal force = 70 kg x 9.8 m/s²
Normal force = 686 N
Now, we can calculate the frictional force:
2. Calculate the frictional force:
Frictional force = coefficient of static friction x Normal force
0.33g = coefficient of static friction x Normal force
Frictional force = 0.33 x 686 N
Frictional force = 226.38 N
The frictional force is equal to the product of the coefficient of static friction and the normal force. Since the normal force and the coefficient of static friction are known, we can solve for the coefficient of static friction:
3. Calculate the coefficient of static friction:
Frictional force = coefficient of static friction x Normal force
226.38 N = coefficient of static friction x 686 N
coefficient of static friction = 226.38 N / 686 N
coefficient of static friction ≈ 0.33
Therefore, the minimum coefficient of static friction required between your feet and the floor to prevent sliding on the train is approximately 0.33.