A crate of oranges weighing 209 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 = 0.41 and μk = 0.20. The truck drives on a straight, level highway at a constant 7.5 m/s.
(a) What is the force of friction acting on the crate?
N
(b) If the truck speeds up with an acceleration of 2.1 m/s2, what is the force of the friction on the crate?
magnitude N
(c) What is the maximum acceleration the truck can have without the crate starting to slide?
m/s^2
I posted this earlier i understand A now which is zero but part B and C are still confusing I couldn't get those please help me thnks
To find the force of friction acting on the crate, we first need to calculate the normal force exerted on the crate by the flatbed truck.
The weight of the crate is given as 209 N. Since the crate is resting on a flat surface, the normal force will be equal to the weight of the crate. Therefore, the normal force is 209 N.
(a) Now that we have the normal force, we can calculate the force of static friction (Fs) using the equation Fs = μs * N, where μs is the coefficient of static friction and N is the normal force.
Fs = 0.41 * 209 N = 85.69 N.
Therefore, the force of friction acting on the crate is 85.69 N.
(b) If the truck speeds up with an acceleration of 2.1 m/s^2, the force of friction acting on the crate will now be the force of kinetic friction (Fk). The equation for kinetic friction is Fk = μk * N, where μk is the coefficient of kinetic friction and N is the normal force.
Fk = 0.20 * 209 N = 41.8 N.
Therefore, the force of friction on the crate when the truck speeds up with an acceleration of 2.1 m/s^2 is 41.8 N.
(c) To find the maximum acceleration the truck can have without the crate starting to slide, we use the equation of motion for the crate:
Net force = m * a,
where m is the mass of the crate and a is its acceleration.
From the given information, we can convert the weight of the crate (209 N) into its mass (m) using the equation:
Weight = m * g,
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
m = Weight / g = 209 N / 9.8 m/s^2 = 21.33 kg.
Now, we can calculate the maximum force of static friction (Fs_max) using the equation Fs_max = μs * N:
Fs_max = 0.41 * 209 N = 85.69 N.
Since the maximum force of static friction is equal to the applied force (since it is the maximum before the crate starts to slide), we can substitute this value into the equation of motion:
Fs_max = m * a,
85.69 N = 21.33 kg * a.
Solving for a, we get:
a = 85.69 N / 21.33 kg ≈ 4.02 m/s^2.
Therefore, the maximum acceleration the truck can have without the crate starting to slide is approximately 4.02 m/s^2.
To solve part B and C of the problem, we need to consider the different cases of motion that can occur:
Part B - When the truck speeds up with an acceleration of 2.1 m/s^2:
In this case, the force of friction between the crate and the truck bed will be kinetic friction because the crate is already in motion. The formula for kinetic friction is given by:
fk = μk * N
where fk is the force of kinetic friction and N is the normal force.
To find the normal force, we need to consider the forces acting on the crate in the vertical direction. Since the crate is at rest vertically, the normal force would be equal to the weight of the crate. The weight of the crate can be calculated using the formula:
weight = mass * gravity
where mass is the mass of the crate and gravity is the acceleration due to gravity (approximately 9.8 m/s^2).
Now let's calculate the force of kinetic friction:
fk = μk * N
Since N = weight, we can substitute:
fk = μk * weight
To find the weight of the crate, we can use the formula:
weight = mass * gravity
Once we have the weight of the crate, we can substitute it into the equation for fk to get the force of kinetic friction.
Finally, the magnitude of the force of friction can be found by multiplying the coefficient of kinetic friction by the weight of the crate.
Part C - Maximum acceleration without sliding:
In this case, we are looking for the maximum acceleration the truck can have without the crate starting to slide. This means that the force of friction between the crate and the truck bed will be static friction because the crate is still at rest.
The formula for static friction is given by:
fs ≤ μs * N
where fs is the force of static friction and N is the normal force.
Similar to part B, we can find the normal force by calculating the weight of the crate. Once we have the weight, we can substitute it into the equation for fs to get the force of static friction.
To find the maximum acceleration, we need to equate the force of static friction to the maximum value it can have, which is μs * N. Using Newton's second law, we can set up the following equation:
fs = mass * acceleration
where mass is the mass of the crate and acceleration is the acceleration of the truck.
By substituting the equation for fs into the equation for mass * acceleration, we can solve for the maximum acceleration the truck can have without the crate starting to slide.
I hope this explanation helps you understand the process of solving parts B and C of the problem. If you have any further questions, please let me know!