A glass of orange juice is on the floor of a subway car traveling along a straight path at constant velocity. Everything's fine. The coefficient of static friction between the glass and the floor is 0.32. The subway suddenly accelerates forward. What is the maximum acceleration that the subway car can have without the orange juice sliding backward on the floor.
2.91
To determine the maximum acceleration that the subway car can have without the orange juice sliding backward, we need to consider the force required to overcome static friction between the glass and the floor.
1. Identify the forces acting on the orange juice:
- The force of gravity acting vertically downward.
- The normal force exerted by the floor in the opposite direction to the force of gravity.
- The static friction force between the glass and the floor.
2. Calculate the maximum static friction force:
The maximum static friction force can be determined using the formula: F_max = μ_s * N, where μ_s is the coefficient of static friction and N is the normal force.
In this case, the normal force is equal to the force of gravity acting on the orange juice since the glass is not accelerating vertically. Therefore, N = m * g, where m is the mass of the orange juice and g is the acceleration due to gravity.
3. Determine the maximum acceleration:
Since the orange juice is on the verge of sliding backward, the static friction force must equal the force required to accelerate the orange juice forward without sliding. Therefore, the maximum static friction force is equal to the mass of the orange juice multiplied by the desired acceleration: F_max = m * a_max.
Combining this with the previous equation (F_max = μ_s * N), we can write: μ_s * N = m * a_max.
We can substitute N with m * g and solve for a_max: μ_s * m * g = m * a_max.
Simplifying the equation, we get: a_max = μ_s * g.
4. Substitute the given values and calculate the maximum acceleration:
Given that the coefficient of static friction, μ_s, is 0.32 and the acceleration due to gravity, g, is approximately 9.8 m/s^2, we can substitute these values into the equation to calculate a_max.
a_max = μ_s * g = 0.32 * 9.8 ≈ 3.136 m/s^2.
Therefore, the maximum acceleration that the subway car can have without the orange juice sliding backward on the floor is approximately 3.136 m/s^2.