A train consists of 50 cars, each of which has a mass of 7.2 × 103 kg. The train has an acceleration of +7.2 × 10-2 m/s2. Ignore friction and determine the tension in the coupling (a) between the 30th and 31st cars and (b) between the 49th and 50th cars.
F=ma
so the tension in a coupling depends on the mass behind the coupling
In the case of the just last car,
tension=masscar*a
now between the 30and 31 car, there are 20 cars behind
tension=20(masscar)a
To determine the tension in the coupling between the 30th and 31st cars, we can use Newton's second law, which states that the force acting on an object is equal to its mass multiplied by its acceleration:
F = m * a
The mass of each car is given as 7.2 × 10^3 kg, and the acceleration of the train is given as +7.2 × 10^-2 m/s^2. Since the 30th and 31st cars are part of the same train, their masses can be added together.
(a) Tension between the 30th and 31st cars:
The total mass of the cars from 1st to 30th is given by M1 = 30 * 7.2 × 10^3 kg
The total mass of the cars from 1st to 31st is given by M2 = 31 * 7.2 × 10^3 kg
The force acting on the 30th car is F1 = M1 * a
The force acting on the 31st car is F2 = M2 * a
However, since the tension in the coupling is the same for both cars, we can write:
F1 + T = F2
Substituting F1 and F2, we get:
M1 * a + T = M2 * a
Rearranging the equation to solve for T, we have:
T = (M2 - M1) * a
Now we can calculate the tension:
T = (31 * 7.2 × 10^3 kg - 30 * 7.2 × 10^3 kg) * 7.2 × 10^-2 m/s^2
Simplifying the equation:
T = 7.2 × 10^3 kg * 7.2 × 10^-2 m/s^2
T = 518.4 N
Therefore, the tension in the coupling between the 30th and 31st cars is 518.4 N.
(b) Tension between the 49th and 50th cars:
Using the same logic and equations as before:
T = (50 * 7.2 × 10^3 kg - 49 * 7.2 × 10^3 kg) * 7.2 × 10^-2 m/s^2
T = 7.2 × 10^3 kg * 7.2 × 10^-2 m/s^2
T = 518.4 N
Therefore, the tension in the coupling between the 49th and 50th cars is also 518.4 N.
To determine the tension in the coupling between the 30th and 31st cars, you need to consider the forces acting on the system.
Firstly, we need to calculate the mass of the cars that are before and after the 30th and 31st cars.
The mass of the cars before the 30th car (including the 30th car) would be:
Mass_before = (30 cars) * (7.2 × 10^3 kg/car)
The mass of the cars after the 31st car would be:
Mass_after = (20 cars) * (7.2 × 10^3 kg/car)
Now, we can calculate the total mass of the system:
Total_mass = Mass_before + Mass_after
Next, we can calculate the total force acting on the system using Newton's second law:
Force = Total_mass * Acceleration
Since the tension force acts in opposite direction to the acceleration, the tension force can be written as:
Tension_30_31 = Force
So, to find the tension in the coupling between the 30th and 31st cars, you need to calculate the total force acting on the system.
Similarly, to find the tension in the coupling between the 49th and 50th cars, you will follow the same steps mentioned above, but this time considering the relevant number of cars before and after the 49th and 50th cars.
Remember to use the same acceleration value for both calculations.