A railroad flatcar is loaded with crates having a coefficient of static friction of 0.394 with the floor. If the train is moving at 57.1 km/h, in how short a distance can the train be stopped at constant acceleration without causing the crates to slide?
Vo = 57.1km/h = 57100m/3600s = 15.9 m/s.
Fc = m*g = Force of the crate.
Fs = u*mg = 0.394mg
Fap-Fs = m*a
Fap-0.394mg = m*0 = 0
Fap = 0.394mg = Force applied.
a=Fap/m = 0.394mg/m=0.394g=0.394*(-9.8)
= -3.86 m/s^2.
V^2 = Vo + 2a*d
d = (V^2-Vo^2)/2a = 0-(15.9^2)/-7.72 =
32.7 m.
To determine the shortest stopping distance of the train without causing the crates to slide, we need to calculate the maximum acceleration that can be applied before the crates start sliding.
Here's how you can approach the problem step by step:
1. Convert the train's velocity from km/h to m/s since the coefficient of friction is typically given in terms of SI units (meters per second).
Conversion factor: 1 km/h = 0.2778 m/s
Speed of train (v) = 57.1 km/h * 0.2778 m/s = 15.86038 m/s
2. The force of static friction (F_friction) acting between the crates and the flatcar can be calculated using the equation:
F_friction = coefficient of static friction * normal force
Since the crates are on a flat surface, the normal force is equal to the weight of the crates (mg), where m is the mass of the crates and g is the acceleration due to gravity (9.8 m/s^2).
3. Assume the crates are point-like or evenly distributed on the flatcar, so the normal force can be calculated as follows:
Normal force = (mass of crates / total number of crates) * g
4. Determine the acceleration needed to stop the train, assuming it stops in the shortest distance.
Acceleration (a) = final velocity^2 / (2 * stopping distance)
Since the final velocity is zero (the train is stopping), the equation simplifies to:
Acceleration (a) = 0 / (2 * stopping distance) = 0
5. The maximum force of static friction that can be applied without causing the crates to slide is equal to the mass of the crates multiplied by the acceleration:
F_friction = mass of crates * acceleration
6. Equating the force of static friction to the maximum force of static friction gives us:
mass of crates * acceleration = coefficient of static friction * normal force
7. Rearrange the equation to solve for the stopping distance:
Stopping distance = mass of crates * acceleration / (coefficient of static friction * normal force)
By plugging in the values for the mass of the crates, coefficient of static friction, and normal force, you can calculate the stopping distance in meters.