A train engine pulls out of a station along a straight horizontal track with five identical freight cars behind it, each of which weighs 88.0 kN. The train reaches a speed of 13.0 m/s within 6.50 min of starting out. Assuming the engine pulls with a constant force during this interval, and ignore air resistance and friction on the freight cars. Find the tension in the coupling between cars 2 and 3. In kN ???
To find the tension in the coupling between cars 2 and 3, we can 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.
First, let's calculate the mass of each freight car. Since we know the weight of each car is 88.0 kN, we can divide this by the acceleration due to gravity (9.8 m/s^2) to find the mass:
Mass = Weight / Acceleration due to gravity
Mass = 88.0 kN / 9.8 m/s^2
Mass = 8,979.59 kg
Since all the freight cars are identical, they each have the same mass of 8,979.59 kg.
Next, we need to find the total force acting on the train. We know the acceleration of the train and the combined mass of the engine and the five freight cars. The equation to calculate force is:
Force = Mass × Acceleration
The train starts from rest and reaches a final speed of 13.0 m/s within 6.50 minutes (or 390 seconds). We can convert this into acceleration using the following equation:
Acceleration = Change in velocity / Time taken
Acceleration = (Final velocity - Initial velocity) / Time taken
Acceleration = (13.0 m/s - 0) / 390 s
Acceleration = 0.033 m/s^2
Now, we can calculate the force:
Force = (Mass of engine + Mass of freight cars) × Acceleration
Force = (Mass of engine + (Mass of one freight car × 5)) × Acceleration
Force = (Mass of engine + 8,979.59 kg × 5) × 0.033 m/s^2
Since the engine is pulling with a constant force, the force pulling cars 2 and 3 is the same as the total force acting on the train.
Finally, we can calculate the tension in the coupling between cars 2 and 3. Since there are five cars in the train, the total force is divided equally among the cars. However, the tension between cars 2 and 3 only acts on these two cars. Therefore, we divide the total force by 5 and multiply it by 2 to get the tension between cars 2 and 3:
Tension = (Force / 5) × 2
Tension = (Force / 5) × 2
Tension = (Force / 5) × 2
Now, substituting the value of Force, we get:
Tension = ((Mass of engine + 8,979.59 kg × 5) × 0.033 m/s^2 / 5) × 2
Simplifying and evaluating, we can find the tension in the coupling between cars 2 and 3 in kilonewtons.