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/s2
To find the force of friction acting on the crate, we need to use the formula:
Force of friction = coefficient of friction * normal force
(a) Given that the weight of the crate is 209 N and the crate is on a flatbed truck, the normal force is equal to the weight of the crate, which is 209 N.
Using the coefficient of static friction (μs = 0.41), we can calculate the force of friction:
Force of friction = 0.41 * 209 N = 85.69 N
So, the force of friction acting on the crate is 85.69 N.
(b) When the truck speeds up with an acceleration of 2.1 m/s^2, the crate will experience a net force in the forward direction due to the truck's acceleration. This force will be a combination of the force of friction acting on the crate and the force pushing the crate forward.
The force pushing the crate forward is equal to the product of the crate's mass and acceleration:
Force pushing crate forward = mass of crate * acceleration
To calculate the mass of the crate, we can use the formula:
Weight = mass * acceleration due to gravity
Given that the weight of the crate is 209 N and the acceleration due to gravity is 9.8 m/s^2, we can solve for the mass:
mass of crate = 209 N / 9.8 m/s^2 = 21.33 kg
Now, we can calculate the force pushing the crate forward:
Force pushing crate forward = 21.33 kg * 2.1 m/s^2 = 44.79 N
To find the force of friction on the crate, we need to subtract the force pushing the crate forward from the force of friction calculated in part (a):
Force of friction = 85.69 N - 44.79 N = 40.90 N
So, the force of friction on the crate when the truck speeds up with an acceleration of 2.1 m/s^2 is 40.90 N.
(c) The maximum acceleration the truck can have without the crate starting to slide is determined by the coefficient of static friction (μs).
The maximum static friction force is given by:
Maximum static friction force = μs * normal force
In this case, the normal force is still equal to the weight of the crate, which is 209 N.
Using the coefficient of static friction (μs = 0.41), we can calculate the maximum static friction force:
Maximum static friction force = 0.41 * 209 N = 85.69 N
Since the maximum static friction force is equal to the force of friction calculated in part (a), the maximum acceleration the truck can have without the crate starting to slide is 0.
To answer these questions, we will use the concepts of friction and Newton's laws of motion.
(a) To calculate the force of friction acting on the crate when it is at rest, we need to determine the maximum static friction. The formula for maximum static friction is:
fs = μs * N
where fs is the force of static friction, and N is the normal force.
The normal force, in this case, is equal to the weight of the crate, which is given as 209 N. Therefore, N = 209 N.
Plugging in the values, the force of static friction can be calculated as:
fs = 0.41 * 209 N ≈ 85.69 N
Hence, the force of friction acting on the crate is approximately 85.69 N.
(b) When the truck speeds up with an acceleration of 2.1 m/s^2, we need to consider kinetic friction. The formula for kinetic friction is:
fk = μk * N
where fk is the force of kinetic friction.
The normal force, N, is still equal to the weight of the crate, which is 209 N.
Plugging in the values, the force of kinetic friction can be calculated as:
fk = 0.20 * 209 N ≈ 41.8 N
Hence, the force of friction acting on the crate when the truck speeds up is approximately 41.8 N.
(c) To find the maximum acceleration the truck can have without the crate starting to slide, we need to consider the maximum static friction. The formula for maximum static friction is the same as mentioned in part (a):
fs = μs * N
We know that fs = 85.69 N and N = 209 N.
Plugging in the values in the formula above and solving for acceleration:
85.69 N = 0.41 * 209 N
Dividing both sides by 0.41:
209 N * a_max = 85.69 N
a_max ≈ 85.69 N / 209 N
a_max ≈ 0.411 m/s^2
Therefore, the maximum acceleration the truck can have without the crate starting to slide is approximately 0.411 m/s^2.