a crate sits on the floor of a boxcar. the coefficient of fritcion between the crate and the florr is .600. what is the maximum speed that the boxcar can go around a curve of radius 200 m. without causing the crate to slide?
The maximum speed that the boxcar can go around a curve of radius 200 m without causing the crate to slide is 8.66 m/s. This can be calculated using the equation v = √(μgR), where μ is the coefficient of friction, g is the acceleration due to gravity (9.8 m/s2), and R is the radius of the curve (200 m).
To determine the maximum speed at which the boxcar can go around the curve without causing the crate to slide, we need to calculate the maximum centripetal force that the crate can experience without exceeding the force of static friction.
The maximum centripetal force is given by the equation:
F_c = m * v^2 / r
Where:
F_c is the centripetal force
m is the mass of the crate
v is the velocity of the boxcar
r is the radius of the curve
The maximum force of static friction is given by:
F_f = μ * N
Where:
F_f is the force of friction
μ is the coefficient of friction
N is the normal force
Assuming that the crate is on a level surface in the boxcar, the normal force (N) is equal to the weight of the crate (m * g), where g is the acceleration due to gravity.
Setting the two equations equal to each other, we have:
F_c = F_f
m * v^2 / r = μ * m * g
Simplifying, we get:
v^2 = μ * r * g
Now, we can plug in the given values:
μ = 0.600 (coefficient of friction)
r = 200 m (radius of the curve)
g = 9.8 m/s^2 (acceleration due to gravity)
Calculating the maximum velocity:
v^2 = 0.600 * 200 * 9.8
v^2 = 1176
v ≈ √1176
v ≈ 34.28 m/s
Therefore, the maximum speed that the boxcar can go around the curve without causing the crate to slide is approximately 34.28 m/s.
To find the maximum speed at which the boxcar can go around a curve without causing the crate to slide, we need to take into account the forces acting on the crate. The two main forces are the gravitational force (mg) and the frictional force (μN), where μ is the coefficient of friction and N is the normal force.
The normal force (N) acting on the crate is equal to the weight (mg) of the crate, since it is sitting on a horizontal floor and there is no vertical acceleration. Therefore, N = mg.
The maximum frictional force (Ff) that can be exerted on the crate without causing it to slide is given by the equation Ff = μN.
In this case, the maximum frictional force is equal to the centripetal force (Fc) required to keep the crate moving in a circle of radius 200 m. So, we have Fc = Ff.
The centripetal force (Fc) is given by the equation Fc = mv^2/r, where m is the mass of the crate, v is the velocity, and r is the radius of the curve.
Setting Fc = Ff, we get mv^2/r = μN.
Substituting N = mg, we have mv^2/r = μmg.
Rearranging the equation, we get v^2 = μgr.
Taking the square root of both sides, we have v = √(μgr).
Now we can plug in the values. Given that the coefficient of friction (μ) is 0.600 and the radius of the curve (r) is 200 m, and we assume the acceleration due to gravity (g) is 9.8 m/s^2:
v = √(0.600 * 9.8 * 200)
v = √(1176)
v ≈ 34.28 m/s
Therefore, the maximum speed that the boxcar can go around the curve without causing the crate to slide is approximately 34.28 m/s.