A crate of oranges weighing 164 N rests on a flatbed truck 2.0 m from the back of the truck. The coefficients of friction between the crate and the bed are μs = 0.30 and μk = 0.20. The truck drives on a straight, level highway at a constant 7.6 m/s
a) What is the force of friction acting on the crate?
b) If the truck speeds up with an acceleration of 1.6 m/s2, what is the force of the friction on the crate?
What is the maximum acceleration the truck can have without the crate starting to slide?
To solve these problems, we need to use the equations related to friction and forces. Let's go through each part step-by-step.
a) What is the force of friction acting on the crate?
The force of static friction (fs) can be calculated using the formula:
fs = μs * N
where μs is the coefficient of static friction and N is the normal force.
Since the crate is on a flatbed truck without any vertical acceleration, the normal force N is equal to the weight of the crate, which is the force due to gravity:
N = mg
where m is the mass and g is the acceleration due to gravity.
Given that the weight of the crate is 164 N, we can calculate the normal force:
N = 164 N
Now, let's calculate the force of static friction:
fs = μs * N
fs = 0.30 * 164 N
fs = 49.2 N
Therefore, the force of friction acting on the crate is 49.2 N.
b) If the truck speeds up with an acceleration of 1.6 m/s2, what is the force of friction on the crate?
In this case, the crate experiences both static and kinetic friction.
To determine whether the crate is still in the static friction state or has transitioned to kinetic friction, we need to compare the maximum static friction force (fs) with the applied force (F).
The maximum static friction force (fs) can be calculated using the same formula as before:
fs = μs * N
Now, let's calculate the maximum static friction force:
fs = 0.30 * 164 N
fs = 49.2 N
Since the applied force is not given, we can calculate it using Newton's second law:
F = m * a
where m is the mass of the crate and a is the acceleration.
To find the mass, we use the formula for weight:
mg = 164 N
Now, let's isolate m:
m = 164 N / g
m = 164 N / 9.8 m/s^2
m ≈ 16.73 kg
Now, let's calculate the applied force:
F = m * a
F = 16.73 kg * 1.6 m/s^2
F ≈ 26.77 N
Comparing the applied force (F) with the maximum static friction force (fs), we can determine if the crate transitions to kinetic friction or remains in static friction.
Since F (26.77 N) is less than fs (49.2 N), the crate remains in the static friction state. Therefore, the force of friction on the crate is still 49.2 N.
c) What is the maximum acceleration the truck can have without the crate starting to slide?
To find the maximum acceleration without the crate starting to slide, we need to compare the applied force (F) with the maximum static friction force (fs).
If the applied force (F) exceeds the maximum static friction force (fs), the crate will start to slide, and kinetic friction will come into play.
In this case, the maximum static friction force is still 49.2 N.
Therefore, for the crate not to start sliding, the applied force (F) should be equal to or less than the maximum static friction force (fs):
F ≤ fs
Let's calculate the maximum applied force:
F = μs * N
F = 0.30 * 164 N
F = 49.2 N
Therefore, the maximum acceleration the truck can have without the crate starting to slide is when the applied force is equal to the maximum static friction force, which is 49.2 N.
To answer these questions, we need to use the equations of motion and the concepts of friction.
a) The force of friction acting on the crate can be found using the equation:
Frictional Force (Ff) = μ * Normal Force
where:
- μ is the coefficient of friction
- Normal Force is the force exerted on the crate perpendicular to the surface.
In this case, the Normal Force is equal to the weight of the crate, since the crate is resting on a flat surface. So, the Normal Force can be calculated as:
Normal Force (N) = Mass * Gravitational Acceleration
where:
- Mass is the mass of the crate
- Gravitational Acceleration is the acceleration due to gravity (approximately 9.8 m/s²).
Given that the weight of the crate (Force due to gravity) is 164 N, we can calculate the Normal Force:
N = 164 N
Now, we can calculate the force of friction by substituting μs into the equation:
Ff = μs * N
Ff = 0.30 * 164 N
Calculating this will give us the answer to part (a).
b) If the truck speeds up with an acceleration of 1.6 m/s², the force of friction acting on the crate will be different. To determine the force of friction in this scenario, we need to consider both the static and kinetic coefficients of friction.
The static friction will be in effect as long as the applied force is less than or equal to the maximum static friction force (μs * N). Once the applied force exceeds this value, the crate will start to slide, and the kinetic friction force will come into play.
If the acceleration is 1.6 m/s², the additional force acting on the crate can be calculated as:
Force (F) = Mass * Acceleration
where:
- Mass is the mass of the crate
Now we can determine if the maximum static friction force is enough to oppose this applied force. If it is, the crate will not slide, and the force of friction will be equal to the applied force (F). However, if the applied force exceeds the maximum static friction force, the crate will start to slide, and the force of friction will be the kinetic friction force (μk * N).
To find the maximum static friction force, we use the equation:
Maximum Static Friction Force (Fsf) = μs * N
Substituting the values for μs and N, we can calculate Fsf.
Now, if F is less than or equal to Fsf, the force of friction on the crate will be F. However, if F exceeds Fsf, the friction force on the crate will be the kinetic friction force, which can be calculated using:
Force of Kinetic Friction (Ffk) = μk * N
Calculating this will give us the answer to part (b).
c) To determine the maximum acceleration the truck can have without the crate starting to slide, we need to find the point where the applied force just equals the maximum static friction force. In other words, when the crate is on the verge of sliding.
So, when the applied force is equal to the maximum static friction force, we have:
Applied Force = Fsf
Using the equation for force (F = Mass * Acceleration), we can substitute Fsf into it and solve for acceleration:
Mass * Acceleration = Fsf
Simplifying for acceleration, we have:
Acceleration = Fsf / Mass
Substituting the values for Fsf and Mass, we can calculate the maximum acceleration.
Calculating this will give us the answer to part (c).