A crate of mass 59.3 kg is being transported on the flatbed of a pickup truck. The coefficient of static friction between the crate and the trucks flatbed is 0.357, and the coefficient of kinetic friction is 0.304.
(a) The truck accelerates forward on level ground. What is the maximum acceleration the truck can have so that the crate does not slide relative to the trucks flatbed?
(b) The truck barely exceeds this acceleration and then moves with constant acceleration, with the crate sliding along its bed. What is the acceleration of the crate relative to the ground?
To find the maximum acceleration of the truck, we need to determine the maximum static friction force that can be exerted on the crate. The formula for static friction force is:
F_static = μ_static * N
Where:
F_static is the static friction force
μ_static is the coefficient of static friction
N is the normal force
(a) Maximum acceleration without sliding:
To find the normal force, we need to consider the forces acting on the crate in the vertical direction. Since the crate is on level ground and not accelerating vertically, the normal force is equal to the weight of the crate, which is given by:
N = m * g
Where:
m is the mass of the crate
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Substituting the given values:
N = 59.3 kg * 9.8 m/s^2
N = 581.74 N
Now, we can calculate the maximum static friction force:
F_static = μ_static * N
F_static = 0.357 * 581.74 N
F_static = 207.72 N
The maximum static friction force is 207.72 N. The maximum acceleration the truck can have without the crate sliding relative to the truck's flatbed is equal to the force divided by the mass of the crate:
a_max = F_static / m
a_max = 207.72 N / 59.3 kg
a_max = 3.50 m/s^2
Therefore, the maximum acceleration the truck can have so that the crate does not slide relative to the truck's flatbed is 3.50 m/s^2.
(b) Acceleration of the crate relative to the ground:
Once the crate starts sliding along the truck's flatbed, the kinetic friction force comes into play. The formula for kinetic friction force is:
F_kinetic = μ_kinetic * N
Where:
F_kinetic is the kinetic friction force
μ_kinetic is the coefficient of kinetic friction
N is the normal force
Since the crate is sliding relative to the truck's flatbed, the kinetic friction force will be acting in the opposite direction of the truck's acceleration. Therefore, the net force acting on the crate along the horizontal direction is:
net force = F_applied - F_kinetic
Where:
F_applied is the applied force (equal to the product of the mass and acceleration)
Since the crate is moving with constant acceleration, the net force must be equal to the product of the mass and acceleration:
net force = m * a
Therefore, we can equate the two equations:
m * a = F_applied - F_kinetic
Rearranging the equation:
F_kinetic = F_applied - m * a
Substituting the given values:
F_kinetic = m * a - μ_kinetic * N
F_kinetic = 59.3 kg * a - 0.304 * 581.74 N
F_kinetic = 59.3 kg * a - 177.07 N
The kinetic friction force is also equal to the mass times the acceleration:
F_kinetic = m * a
Equating the two equations:
59.3 kg * a - 177.07 N = 59.3 kg * a
Simplifying the equation:
-177.07 N = 0
This equation cannot be satisfied because it results in a contradiction. Therefore, the crate cannot move with constant acceleration relative to the ground.