Two crates, of mass m1 = 60 kg and m2 = 130 kg, are in contact and at rest on a horizontal surface (Fig. 4-54). A 620 N force is exerted on the 60 kg crate. The coefficient of kinetic friction is 0.15.
(a) Calculate the acceleration of the system.(to the right)
(b) Calculate the force that each crate exerts on the other.
(c) Repeat with the crates reversed.
Find acceleration and crate force.
To solve this problem, we will use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
(a) To calculate the acceleration of the system, we need to determine the net force acting on the crates. This can be found by subtracting the force of friction from the applied force.
The force of friction can be calculated using the equation:
Force of friction = coefficient of kinetic friction * normal force
The normal force is equal to the weight of the crates, which can be calculated as:
Weight = mass * gravity
where gravity is approximately 9.8 m/s^2.
For the 60 kg crate:
Weight = 60 kg * 9.8 m/s^2 = 588 N
For the 130 kg crate:
Weight = 130 kg * 9.8 m/s^2 = 1,274 N
Now, let's calculate the force of friction for each crate:
Force of friction for the 60 kg crate = 0.15 * 588 N = 88.2 N
Force of friction for the 130 kg crate = 0.15 * 1,274 N = 191.1 N
The net force acting on the system is equal to the applied force minus the force of friction on the 60 kg crate:
Net force = Applied force - Force of friction for the 60 kg crate
Net force = 620 N - 88.2 N = 531.8 N
To calculate the acceleration of the system, we can use Newton's second law:
Net force = (m1 + m2) * acceleration
where m1 and m2 are the masses of the crates.
Acceleration = Net force / (m1 + m2)
Acceleration = 531.8 N / (60 kg + 130 kg)
Acceleration = 531.8 N / 190 kg
Acceleration ≈ 2.80 m/s^2
Therefore, the acceleration of the system is approximately 2.80 m/s^2 to the right.
(b) To calculate the force that each crate exerts on the other, we need to consider that the force applied to the 60 kg crate is also acting on the 130 kg crate.
Force exerted by the 60 kg crate on the 130 kg crate = applied force = 620 N
Force exerted by the 130 kg crate on the 60 kg crate = the force of friction for the 130 kg crate = 191.1 N
Therefore, the force that each crate exerts on the other is 620 N and 191.1 N respectively.
(c) When the crates are reversed, the masses of the crates are interchanged.
For the 60 kg crate:
Weight = 60 kg * 9.8 m/s^2 = 588 N
For the 130 kg crate:
Weight = 130 kg * 9.8 m/s^2 = 1,274 N
The force of friction for each crate remains the same:
Force of friction for the 60 kg crate = 0.15 * 588 N = 88.2 N
Force of friction for the 130 kg crate = 0.15 * 1,274 N = 191.1 N
Now, let's calculate the net force and the acceleration of the system:
Net force = applied force - force of friction for the 130 kg crate
Net force = 620 N - 191.1 N = 428.9 N
Acceleration = Net force / (m1 + m2)
Acceleration = 428.9 N / (130 kg + 60 kg)
Acceleration = 428.9 N / 190 kg
Acceleration ≈ 2.26 m/s^2
Therefore, the acceleration of the system is approximately 2.26 m/s^2 to the right when the crates are reversed.
To calculate the force that each crate exerts on the other, we consider that the force applied to the 130 kg crate is acting on the 60 kg crate.
Force exerted by the 130 kg crate on the 60 kg crate = applied force = 620 N
Force exerted by the 60 kg crate on the 130 kg crate = the force of friction for the 60 kg crate = 88.2 N
Therefore, when the crates are reversed, the force that each crate exerts on the other is 88.2 N and 620 N respectively.
To calculate the acceleration of the system and the force that each crate exerts on the other, we can use Newton's second law of motion and the laws of friction.
(a) To find the acceleration of the system, we need to calculate the net force acting on the system. The net force is the difference between the applied force and the frictional force.
1. Calculate the frictional force:
Frictional force = coefficient of kinetic friction * normal force
The normal force is the force exerted by the surface on the crates, which is equal to the weight of the crates:
Normal force = (mass of the 60 kg crate + mass of the 130 kg crate) * acceleration due to gravity
Normal force = (60 kg + 130 kg) * 9.8 m/s^2
2. Calculate the frictional force:
Frictional force = 0.15 * Normal force
3. Calculate the net force:
Net force = applied force - frictional force
Net force = 620 N - frictional force
4. Calculate the acceleration:
Using Newton's second law: Net force = mass * acceleration
620 N - frictional force = (mass of the 60 kg crate + mass of the 130 kg crate) * acceleration
Solving this equation will give you the acceleration of the system.
(b) To calculate the force that each crate exerts on the other, we need to consider the interaction between the two crates. Since the two crates are in contact and at rest, the force exerted by one crate on the other is equal in magnitude and opposite in direction.
Therefore, the force exerted by the 60 kg crate on the 130 kg crate is equal in magnitude and opposite in direction to the force exerted by the 130 kg crate on the 60 kg crate.
The force exerted by each crate on the other can be calculated using Newton's third law of motion.
(c) To repeat the calculation with the crates reversed, you need to swap the masses in all the calculations. Calculate the new acceleration and the force that each crate exerts on the other using the same steps mentioned above, but with the masses of the crates reversed (60 kg becomes 130 kg and vice versa).