three crates are connected by a string and pulley system. The mass of the crate hanging on the left side of the table is 0.820 kg and the mass of the crate on the table is 1.78 kg. If the coefficient of kinetic friction between the table and the 1.78 kg crate is 0.570, what is the acceleration of the 1.78 kg crate? Assume the string and pulleys are massless.
To find the acceleration of the 1.78 kg crate, we need to analyze the forces acting on it.
First, let's identify the forces acting on the 1.78 kg crate. We have:
1. Force due to gravity (weight): This force acts vertically downward with a magnitude of (mass × acceleration due to gravity). So, the weight of the 1.78 kg crate is (1.78 kg × 9.8 m/s²) = 17.444 N.
2. Normal force: This force is exerted by the table on the crate in a direction perpendicular to the table surface. It counteracts the force due to gravity.
3. Force of kinetic friction: This force opposes the motion of the crate on the table. The magnitude of frictional force is given by (coefficient of kinetic friction × normal force). Here, the coefficient of kinetic friction is given as 0.570.
Now, to find the normal force, we need to consider the system as a whole. The system consists of the 1.78 kg crate on the table and the 0.82 kg crate hanging on the left side.
The tension in the string is the same throughout the system. So, the tension in the string supports the weight of the hanging crate. Therefore, the normal force exerted by the table on the 1.78 kg crate is equal to the weight of the hanging crate, which is (0.82 kg × 9.8 m/s²) = 8.036 N.
Now, let's calculate the force of kinetic friction. The magnitude of the frictional force is (coefficient of kinetic friction × normal force) = (0.570 × 8.036 N) = 4.58292 N.
Since the frictional force opposes the motion, its direction is opposite to the applied force. Therefore, the net force acting on the 1.78 kg crate is the difference between the weight and the frictional force. So, the net force is (17.444 N - 4.58292 N) = 12.86108 N.
Using Newton's second law (F = m × a), we can calculate the acceleration:
acceleration = net force / mass
acceleration = 12.86108 N / 1.78 kg
acceleration ≈ 7.222 m/s²
Therefore, the acceleration of the 1.78 kg crate is approximately 7.222 m/s².