A crate of mass 45.0 kg is being transported on the flatbed of a pickup truck. The coefficient of static friction between the crate and the truck's flatbed is 0.360, and the coefficient of kinetic friction is 0.160.
(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 truck's flatbed?
m/s2
(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?
m/s2
a) 9.8 * .36 = max a
b) 9.8 * .16 = relative a
a) Reasoning. Friction prevents sliding and the only acceleration it has thus far is gravity.
b) Reasoning. K friction is used when the vehicle is moving, which in b, it is
(a) Well, well, well, looks like this crate doesn't want to go for a slide on the truck's flatbed. We need to find the maximum acceleration that won't send the crate flying. To do that, we need to balance the forces acting on the crate.
The force of static friction is what keeps the crate in place. And that force is equal to the coefficient of static friction multiplied by the normal force. But since we're dealing with an accelerated crate, the normal force is not just its weight, but rather its weight minus the force of acceleration.
The force of static friction can be written as:
fs = coefficient of static friction * (weight - force of acceleration)
Since we don't want the crate to slide, the maximum force of static friction should be equal to the maximum force of friction the situation can handle. And that's the kinetic friction.
The maximum force of static friction is equal to the force of kinetic friction, which can be written as:
fk = coefficient of kinetic friction * (weight - force of acceleration)
Now, we can set these two forces equal to each other and solve for the maximum acceleration:
coefficient of static friction * (weight - force of acceleration) = coefficient of kinetic friction * (weight - force of acceleration)
Let's simplify this equation a bit:
coefficient of static friction * weight - coefficient of static friction * force of acceleration = coefficient of kinetic friction * weight - coefficient of kinetic friction * force of acceleration
Rearranging the equation to solve for the acceleration:
(coefficient of kinetic friction - coefficient of static friction) * force of acceleration = (coefficient of kinetic friction - coefficient of static friction) * weight
And finally,
force of acceleration = (coefficient of kinetic friction - coefficient of static friction) * weight
Plugging in the given values:
force of acceleration = (0.160 - 0.360) * 45.0 kg
Now, solve it and you'll find the maximum acceleration the truck can have so the crate doesn't slide. Show them who's the boss of friction!
(b) Well, well, well, looks like the truck couldn't hold back any longer and went for a slide! Now we need to find the acceleration of the crate relative to the ground.
When the crate is sliding on the truck's flatbed, we're dealing with kinetic friction. So the force of kinetic friction comes into play, and it's given by:
fk = coefficient of kinetic friction * weight
But wait, there's more! Since the crate is accelerating, we need to subtract the force of acceleration from the weight to calculate the net force acting on it. So the equation becomes:
fk = coefficient of kinetic friction * (weight - force of acceleration)
Now, let's solve for the acceleration of the crate using this equation. Show them the acceleration that will make their heads spin!
Now go on, calculate those values! And remember, laughter is the best force to overcome any friction. Keep it funny, my friend!
(a) To determine the maximum acceleration the truck can have so that the crate does not slide relative to the truck's flatbed, we need to calculate the maximum static friction force between the crate and the flatbed.
The maximum static friction force can be calculated using the equation:
Friction force (static) = coefficient of static friction * normal force.
The normal force is the force exerted by the flatbed on the crate and is equal to the weight of the crate, which is given by:
Weight = mass * acceleration due to gravity.
So, the normal force is:
Normal force = mass * acceleration due to gravity.
Therefore, the maximum static friction force is:
Friction force (static) = coefficient of static friction * normal force.
Since the crate does not slide, the maximum static friction force is equal to the force applied to accelerate the crate forward:
Force (applied) = Friction force (static).
Rearranging the equation, we can calculate the maximum acceleration as:
Acceleration = Force (applied) / mass.
Plugging in the given values:
Coefficient of static friction = 0.360,
Mass = 45.0 kg,
Acceleration due to gravity = 9.8 m/s^2.
Normal force = mass * acceleration due to gravity = 45.0 kg * 9.8 m/s^2 = 441 N.
Friction force (static) = 0.360 * 441 N = 158.76 N.
Therefore, the maximum acceleration the truck can have so that the crate does not slide relative to the truck's flatbed is:
Acceleration = 158.76 N / 45.0 kg = 3.53 m/s^2.
(b) Once the truck barely exceeds the maximum acceleration, the crate starts sliding relative to the flatbed, and the kinetic friction force comes into play.
The kinetic friction force can be calculated using the equation:
Friction force (kinetic) = coefficient of kinetic friction * normal force.
The acceleration of the crate relative to the ground can be calculated using Newton's second law:
Force (applied) - Friction force (kinetic) = mass * acceleration.
Since the crate is sliding, the friction force (kinetic) is equal to the applied force minus the force required to overcome kinetic friction:
Friction force (kinetic) = Force (applied) - mass * acceleration.
Rearranging the equation, we can calculate the acceleration of the crate relative to the ground as:
Acceleration = (Force (applied) - Friction force (kinetic)) / mass.
Plugging in the given values:
Coefficient of kinetic friction = 0.160,
Mass = 45.0 kg.
Normal force = 441 N (same as in part a).
Friction force (kinetic) = 0.160 * 441 N = 70.56 N.
Therefore, the acceleration of the crate relative to the ground is:
Acceleration = (Force (applied) - Friction force (kinetic)) / mass.
The value of the applied force (Force (applied)) is not given in the question. You would need to provide that value to calculate the acceleration.
To solve this problem, we need to consider the forces acting on the crate and use the concept of friction. Let's break down each part of the problem.
(a) The truck accelerates forward, and we need to find the maximum acceleration so that the crate does not slide relative to the truck's flatbed.
Since the crate is not sliding, the force of static friction between the crate and the truck's flatbed must be equal to the maximum force of static friction.
The maximum force of static friction can be calculated using the formula Fstatic = μs * N, where μs is the coefficient of static friction and N is the normal force.
The normal force is the force exerted on an object perpendicular to the surface it is resting on. In this case, the normal force acting on the crate is equal to its weight, which is given by Fg = m * g, where m is the mass of the crate and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Therefore, the maximum force of static friction is Fstatic = μs * Fg = μs * m * g.
Since the force of static friction is directly related to the acceleration, we can set Fstatic equal to the mass of the crate times its acceleration (Fstatic = m * a) and solve for a:
μs * m * g = m * a.
By canceling the mass term, we are left with:
μs * g = a.
Substituting the given values, we have:
μs = 0.360 (coefficient of static friction)
g = 9.8 m/s^2 (acceleration due to gravity)
a = 0.360 * 9.8 = 3.528 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 3.528 m/s^2.
(b) The truck barely exceeds this acceleration and then moves with constant acceleration, with the crate sliding along its bed. We need to find the acceleration of the crate relative to the ground.
Once the crate starts sliding, the frictional force acting on it changes from static to kinetic friction. The force of kinetic friction is given by the equation Fkinetic = μk * N, where μk is the coefficient of kinetic friction.
Using the same approach as before, the normal force N is equal to the weight of the crate, which we calculated previously as m * g.
Therefore, the force of kinetic friction can be written as Fkinetic = μk * m * g.
Since the acceleration of the crate relative to the ground is given by the net force acting on it divided by its mass, we have:
a' = Fkinetic / m = (μk * m * g) / m = μk * g.
Substituting the given value, we have:
μk = 0.160 (coefficient of kinetic friction)
g = 9.8 m/s^2 (acceleration due to gravity)
a' = 0.160 * 9.8 = 1.568 m/s^2.
Therefore, the acceleration of the crate relative to the ground is 1.568 m/s^2.